3. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
4. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
5. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule:
Divide–Subtract Rule:
Power–Multiply Rule:
6. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK = AN+K
Divide–Subtract Rule:
Power–Multiply Rule:
7. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK = AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule:
8. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK = AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule: (AN)K = ANK
x9
9. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK = AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule: (AN)K = ANK
9x5 =x14 , x9 = x9–5 = x4, and (x9)5 = x45.
For example, x x5
10. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK =AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule: (AN)K = ANK
9x5 =x14 , x9 = x9–5 = x4, and (x9)5 = x45.
For example, x x5
These particular operation–conversion rules appear often in
other forms in mathematics.
11. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK =AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule: (AN)K = ANK
9x5 =x14 , x9 = x9–5 = x4, and (x9)5 = x45.
For example, x x5
These particular operation–conversion rules appear often in
other forms in mathematics. Hence their names, the Multiply–
Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule,
are important.
12. Review on Exponents
Let’s review the basics of exponential notation.
The quantity A multiplied to itself N times is written as AN.
A x A x A ….x A = AN exponent
N times base
Rules of Exponents
Multiply–Add Rule: ANAK =AN+K
AN = AN – K
Divide–Subtract Rule: AK
Power–Multiply Rule: (AN)K = ANK
9x5 =x14 , x9 = x9–5 = x4, and (x9)5 = x45.
For example, x x5
These particular operation–conversion rules appear often in
other forms in mathematics. Hence their names, the Multiply–
Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule,
are important. Let’s extend the definition to negative and
fractional exponents.
16. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
17. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0
Since AK AK
18. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K,
Since AK AK
19. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
20. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k
Since ( A )k = A = (A1/k )k,
21. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
22. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
23. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
24. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by of the exponent last.
25. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 =
b. 91/2 =
c. 9 –3/2 =
26. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81
b. 91/2 =
c. 9 –3/2 =
27. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81
b. 91/2 = √9 = 3
c. 9 –3/2 =
28. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81
b. 91/2 = √9 = 3
c. 9 –3/2 =
29. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81 Pull the numerator outside to
take the root and simplify the
b. 91/2 = √9 = 3 base first.
c. 9 –3/2 = (9½)–3
30. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81 Pull the numerator outside to
take the root and simplify the
b. 91/2 = √9 = 3 base first.
c. 9 –3/2 = (9½)–3 = 3–3
31. The Exponential Functions
A1
Since A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
0-Power Rule: A0 = 1
1 = A0 = A0 – K = A–K, we get the Negative Power Rule.
Since AK AK
Negative Power Rule: A–K = 1 K A
k k
Since ( A )k =A= (A1/k )k, hence A1/k = A.
k
Fractional Powers: A1/k = A.
For a general fractional exponent, we interpret the operations
step by step by doing the numerator of the exponent last.
Example A. Simplify.
a. 9–2 = 12 = 1
9 81 Pull the numerator outside to
take the root and simplify the
b. 91/2 = √9 = 3 base first.
c. 9 –3/2 = (9½)–3 = 3–3 = 13 = 1
3 27
33. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
34. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8
35. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
36. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
37. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
Rational Power equations are equations of the type xR = c
where R = P/Q is a rational number.
38. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
Rational Power equations are equations of the type xR = c
where R = P/Q is a rational number. To solve them, we take the
reciprocal power,
39. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
Rational Power equations are equations of the type xR = c
where R = P/Q is a rational number. To solve them, we take the
reciprocal power, that is,
if xR = c, or xP/Q = c
40. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
Rational Power equations are equations of the type xR = c
where R = P/Q is a rational number. To solve them, we take the
reciprocal power, that is,
if xR = c, or xP/Q = c Reciprocate the powers
then x = ( )c1/R or x = ( )cQ/P
41. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
3
x = √–8 = –2.
Using fractional exponent, we write these steps as
if x3 = –8 then
The reciprocal of the power 3
x = (–8)1/3 = –2.
Rational Power equations are equations of the type xR = c
where R = P/Q is a rational number. To solve them, we take the
reciprocal power, that is,
if xR = c, or xP/Q = c Reciprocate the powers
then x = ( )c1/R or x = ( )cQ/P
However, depending on the values of c and Q/P, it may be that
there is no real solutions, exactly one real solutions,
or both ( ) c1/R are real solutions.
42. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
43. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
44. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
45. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
46. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
47. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
48. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
49. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
x = (–64)1/2 so there is no real number solution.
50. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
x = (–64)1/2 so there is no real number solution.
d. x –2/3 = 64
51. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
x = (–64)1/2 so there is no real number solution.
d. x –2/3 = 64
x = 64–3/2
x = (√64)–3
52. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
x = (–64)1/2 so there is no real number solution.
d. x –2/3 = 64
x = 64–3/2
x = (√64)–3 = 4–3 = 1/64.
53. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
We note that this is the only solution.
b. x2 = 64
x = 641/2 or that
x = √64 = 8.
We note that both 8 are solutions.
c. x2 = –64
x = (–64)1/2 so there is no real number solution.
d. x –2/3 = 64
x = 64–3/2
x = (√64)–3 = 4–3 = 1/64.
Again we check that both 1/64 are solutions.
54. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
55. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
56. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
57. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2
58. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8.
59. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
60. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
61. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
62. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
4 or
5
Mathematics expressions such 3 3 + x2 that appeared in
books can’t be input “as is” in calculators, smart phone apps or
computer software.
63. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
4 or
5
Mathematics expressions such 3 3 + x2 that appeared in
books can’t be input “as is” in calculators, smart phone apps or
computer software. While there is no standard input method
that would work for all of them, most digital input methods are
similar to the BASIC–input method which is used in the
computer language BASIC.
64. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
4 or
5
Mathematics expressions such 3 3 + x2 that appeared in
books can’t be input “as is” in calculators, smart phone apps or
computer software. While there is no standard input method
that would work for all of them, most digital input methods are
similar to the BASIC–input method which is used in the
computer language BASIC. For example, the book–format 3 4
is “3/4” in the BASIC format. (All inputs will be in “quoted
boldface”.)
65. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
4 or
5
Mathematics expressions such 3 3 + x2 that appeared in
books can’t be input “as is” in calculators, smart phone apps or
computer software. While there is no standard input method
that would work for all of them, most digital input methods are
similar to the BASIC–input method which is used in the
computer language BASIC. For example, the book–format 3 4
is “3/4” in the BASIC format. (All inputs will be in “quoted
boldface”.) The power operation is “^”, shift + 6 on the
standard keyboard.
66. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, we solve for the
power term first, then apply the reciprocal power to find x.
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4
x = 43/2 x = (√4)3 = 8. Note that both 8 are solutions.
For the approximate irrational answers, we need calculators.
Calculator Inputs
4 or
5
Mathematics expressions such 3 3 + x2 that appeared in
books can’t be input “as is” in calculators, smart phone apps or
computer software. While there is no standard input method
that would work for all of them, most digital input methods are
similar to the BASIC–input method which is used in the
computer language BASIC. For example, the book–format 3 4
is “3/4” in the BASIC format. (All inputs will be in “quoted
boldface”.) The power operation is “^”, shift + 6 on the
standard keyboard. Hence the BASIC–input of 34 is “3^4”.
68. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
69. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?).
70. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
71. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
72. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity
for the subtraction
operation
73. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ”
for the subtraction
operation
74. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ”
for the subtraction
operation Some calculators allow skipping
the last parenthesis, but most
would not. It’s an error in BASIC.
75. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ” Missing a quantity
for the subtraction for the multiplication
operation operation
76. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ” Missing a quantity
for the subtraction for the multiplication
operation operation
Following are some of the common syntax errors.
77. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ” Missing a quantity
for the subtraction for the multiplication
operation operation
Following are some of the common syntax errors.
* It take two quantities to perform +, --, *, / and ^.
78. Power Equations and Calculator Inputs
Syntax
The word “syntax” refers to the rules for recognizable input.
If we enter “5 + ” in the calculator, press “enter”, we get an
error message (5 plus what?). Such an error where the input
is incomplete or gibberish is called a “syntax error”.
Example A. What is wrong with each of the following
expressions. Fill in symbols to make them syntactically
correct. (There is more than one way to do that.)
a. (6+(3–)) b. (6–4(0–(–1)) c. 1.3*2.1/(4.3*)^3
Missing a quantity Missing a “ ) ” Missing a quantity
for the subtraction for the multiplication
operation operation
Following are some of the common syntax errors.
* It take two quantities to perform +, --, *, / and ^.
Inputs such as “6*”, “23 /” , or “7^” are not executable.
79. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it.
80. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
81. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
82. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message.
83. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
84. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
Semantics
85. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
Semantics
Semantics are the rules for interpreting the "meaning“ of
syntactically correct expressions.
86. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
Semantics
Semantics are the rules for interpreting the "meaning“ of
syntactically correct expressions. The semantics for BASIC–
format inputs are the same as the rules for order of operations
PEMDAS.
87. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
Semantics
Semantics are the rules for interpreting the "meaning“ of
syntactically correct expressions. The semantics for BASIC–
format inputs are the same as the rules for order of operations
PEMDAS. A semantic input mistake is a mistake where we
mean to execute one set of calculations without realizing that
the input is interpreted differently by the machine.
88. Power Equations and Calculator Inputs
* For every left parentheses “( ” opened , there must be a
right parentheses “)” that closes it. Hence the number of “( ”
should be the same the number of “)” in the inputs.
* Symbols such as [ ] or { } may have different meanings.
The machine always tell us when we made a syntax error with
an error message. However another type of mistake that occurs
often for which there is no warning and is difficult to catch are
the “miscommunications “.
Semantics
Semantics are the rules for interpreting the "meaning“ of
syntactically correct expressions. The semantics for BASIC–
format inputs are the same as the rules for order of operations
PEMDAS. A semantic input mistake is a mistake where we
mean to execute one set of calculations without realizing that
the input is interpreted differently by the machine. Here are
some examples of similar but different inputs.
89. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
c. Find “–2–4^(1/2)/2” with and without the calculator.
d. Write the BASIC input of 2–√4 . Find the answer.
2
90. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 .
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
c. Find “–2–4^(1/2)/2” with and without the calculator.
d. Write the BASIC input of 2–√4 . Find the answer.
2
91. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
c. Find “–2–4^(1/2)/2” with and without the calculator.
d. Write the BASIC input of 2–√4 . Find the answer.
2
92. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
2
The BASIC input of –8 is “–8^(2/3)”.
3
c. Find “–2–4^(1/2)/2” with and without the calculator.
d. Write the BASIC input of 2–√4 . Find the answer.
2
93. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
2
The BASIC input of –8 is “–8^(2/3)”. The answer is –4.
3
c. Find “–2–4^(1/2)/2” with and without the calculator.
The input yields –3 because it’s –2 –√4 .
2
d. Write the BASIC input of 2–√4 . Find the answer.
2
94. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
2
The BASIC input of –8 is “–8^(2/3)”. The answer is –4.
3
c. Find “–2–4^(1/2)/2” with and without the calculator.
The input yields –3 because it’s –2 –√4 .
2
d. Write the BASIC input of 2–√4 . Find the answer.
2
The correct BASIC format is “(–2–4^(1/2))/2”.
95. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
2
The BASIC input of –8 is “–8^(2/3)”. The answer is –4.
3
c. Find “–2–4^(1/2)/2” with and without the calculator.
The input yields –3 because it’s –2 –√4 .
2
d. Write the BASIC input of 2–√4 . Find the answer.
2
The correct BASIC format is “(–2–4^(1/2))/2”. The answer is 0.
96. Power Equations and Calculator Inputs
Example B.
a. Find “–8^2/3” with and without the calculator .
–43 = –64 . The input yields –64/3 = –21.33…
It’s 3 3
2
b. Write –8 in the BASIC input, find the answer with and
3
without the calculator.
2
The BASIC input of –8 is “–8^(2/3)”. The answer is –4.
3
c. Find “–2–4^(1/2)/2” with and without the calculator.
The input yields –3 because it’s –2 –√4 .
2
d. Write the BASIC input of 2–√4 . Find the answer.
2
The correct BASIC format is “(–2–4^(1/2))/2”. The answer is 0.
In general, when in doubt, insert ( )’s to specify the order of
operations.
