The document discusses solving power equations and proper calculator input format for expressions involving powers and fractions. It provides examples of solving various power equations by taking the reciprocal of the power. It also emphasizes the need for precise text input, such as using parentheses and the caret symbol "^", to evaluate expressions correctly on a calculator. Common mistakes like incorrect ordering of operations when inputting a fraction or power are highlighted.

1. Power Equations and Calculator Inputs
Power Equations

2. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is

3. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3

4. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then

5. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3
The reciprocal of the power 3

6. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
The reciprocal of the power 3

7. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
The reciprocal of the power 3

8. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
The reciprocal of the power 3

9. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
then x = (±) c Q/P.
The reciprocal of the power P/Q
The reciprocal of the power 3

10. Power Equations and Calculator Inputs
Power Equations
The solution to the equation
x 3 = –8 is
x = √–8 = –2.
3
Using fractional exponent notation, we write these steps as
if x3 = –8 then
x = (–8)1/3 = –2.
(Rational) Power equations are equations of the type xP/Q = c.
To solve them, we take the reciprocal power, that is,
if xP/Q = c,
then x = (±) c Q/P.
Note that xP/Q may not exist, or that sometime we get both (±)
xP/Q solutions means that sometimes.
The reciprocal of the power P/Q
The reciprocal of the power 3

11. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
b. x2 = 64
c. x2 = –64
d. x –3/2 = 64

12. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3
b. x2 = 64
c. x2 = –64
d. x –3/2 = 64

13. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64
b. x2 = 64
c. x2 = –64
d. x –3/2 = 64

14. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
x = √64 = 4.
b. x2 = 64
c. x2 = –64
d. x –3/2 = 64

15. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
c. x2 = –64
d. x –3/2 = 64

16. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2
We note that both ±8 are solutions.
c. x2 = –64
d. x –3/2 = 64

17. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
d. x –3/2 = 64

18. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2
d. x –3/2 = 64

19. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –3/2 = 64

20. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –3/2 = 64
x = 64–3/2

21. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –3/2 = 64
x = 64–3/2
x = (√64)–3

22. Power Equations and Calculator Inputs
Example A. Solve for the real solutions.
a. x3 = 64
x = 641/3 or that
3
We note that this is the only solution.
x = √64 = 4.
b. x2 = 64
x = 641/2 or that
We note that both ±8 are solutions.
x = √64 = 8.
c. x2 = –64
x = (–64)1/2 which is UDF. (In fact what most calculators
return as the answer meaning that there is no real solutions.)
d. x –3/2 = 64
x = 64–3/2
x = (√64)–3
= 8–3 = 1/512.

23. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, solve for the
power term first,
e. 2x2/3 – 7 = 1

24. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, solve for the
power term first,
e. 2x2/3 – 7 = 1
2x2/3 = 8

25. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, solve for the
power term first,
e. 2x2/3 – 7 = 1
2x2/3 = 8
x2/3 = 4

26. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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

27. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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

28. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.

29. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.

30. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format.

31. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.

32. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”.

33. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”. For example, the
fraction is inputted as “3/4”, and the quantity 34 is “3^4”.3
4

34. Power Equations and Calculator Inputs
Finally, for linear form of the power equations, 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.
We note that both ±8 are solutions.
Mathematics Inputs in Text Format
Most digital calculation devices such as calculators, smart
phone apps or computer software accept inputs in the text
format. Besides the “+” , “–”, for addition and subtraction we
use “ * ” for multiplication, and “/” for the division operation.
The power operation is represented by “^”. For example, the
fraction is inputted as “3/4”, and the quantity 34 is “3^4”.
All executions of such inputs follow the order of operations.
3
4

35. Power Equations and Calculator Inputs
Many common input mistakes happen for expressions involving
division or taking powers.

36. Power Equations and Calculator Inputs
Example B.
a. Input and execute with a calculator.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
3
2
4
b. Input and execute with a calculator.

37. Power Equations and Calculator Inputs
Example B.
a. Input and execute with a calculator.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is 4^(3/2) to get the correct answer of 8.
3
2
4
b. Input and execute with a calculator.

38. Power Equations and Calculator Inputs
Example B.
a. Input and execute with a calculator.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is 4^(3/2) to get the correct answer of 8.
(The incorrect input 4^3/2 gives the answer 43/2 or 32.)
3
2
4
b. Input and execute with a calculator.

39. Power Equations and Calculator Inputs
Example B.
a. Input and execute with a calculator.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is 4^(3/2) to get the correct answer of 8.
(The incorrect input 4^3/2 gives the answer 43/2 or 32.)
3
2
4
b. Input and execute with a calculator.
The correct text input is (2+6)/4^(3/2) for the correct answer 1.

40. Power Equations and Calculator Inputs
Example B.
a. Input and execute with a calculator.
Many common input mistakes happen for expressions involving
division or taking powers.
3
2
4
2 + 6
The correct text input is 4^(3/2) to get the correct answer of 8.
(The incorrect input 4^3/2 gives the answer 43/2 or 32.)
In general, when in doubt, insert ( )’s in the input to clarify the
order of operations.
3
2
4
b. Input and execute with a calculator.
The correct text input is (2+6)/4^(3/2) for the correct answer 1.

41. Power Equations and Calculator Inputs
1. 2x2 = 8 2. x2 = 0.09 3. x2 = –9
Exercise.