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Exponents and
Radicals
By-
Satyansh Sharad, 8th A
Introduction
 an = a * a * a * a…* a (where there are n factors)
 The number ‘a’ is the base and ‘n’ is the exponent.
 For any non-zero rational number a, we define a0 =
1
 Let a be any non-zero rational number and n be a
positive integer, then we define a-n = 1/an
Laws of Exponents
aman = am+n
When multiplying two powers of the same base,
add the exponents.
am/ an = am – n
When dividing two powers of the same base,
subtract the exponents.
(am)n = amn
When raising a power to a power, multiply the
exponents.
Laws of Exponents
(ab)n = anbn
When raising a product to a power, raise each factor to the
power.
(a/b)n = an / bn
When raising a quotient to a power, raise both the numerator
and denominator to the power.
(a/b)-n = (b/a)n
When raising a quotient to a negative power, take the
reciprocal and change the power to a positive.
a-m / b-n = bm / an
To simplify a negative exponent, move it to the opposite
position in the fraction. The exponent then becomes
positive.
nth root
If n is any positive integer, then the principal nth root of a is defined
as:
If n is even, a and b must be positive.
means nn
a b b a 
Properties of nth roots
if n is odd
| | if n is even
n n n
n
n
n
m n mn
n n
n n
ab a b
a a
b b
a a
a a
a a





Rational Exponents
 For any rational exponent m/n in lowest terms, where m
and n are integers and n>0, we define:
If n is even, then we require that a ≥ 0.
 am is called the radicand
 n is called the index
 is called the Radical
 In general, am/n=(am)1/n
 In general, a-m/n= 1/am/n =1/(am)1/n
/ nm n m
a a
/ nm n m
a a
Examples
 Express in exponential form:
• √7
 SOLUTION:
√7 = 7 ½
 Express in radical form:
• (5) 1/3
 SOLUTION:
(5) 1/3 =
𝟑
𝟓
Examples
 Find the value of
• (125) 2/3
 SOLUTION:
(125) 2/3
= (
𝟑
𝟏𝟐𝟓)2
= 52
= 25
Examples
 Find the value of
• (27)-2/3
 SOLUTION:
(27) -2/3
= (
𝟑
𝟐𝟕)-2
= 3-2
= 1/9
Examples
 Find the value of
• (6) ½ x (6) 3/2
 SOLUTION:
(6) ½ + 3/2
= 6 4/2
= 62
= 36
Examples
 Simplify
• (27)6/5 /(27) 1/5
 SOLUTION:
(27) 6/5- 1/5
=(27)5/5
=27
Examples
 Evaluate
• (0.000064) 5/6
 SOLUTION:
(64/1000000) 5/6
= [(2/10)6]5/6
=(2/10)5
= (32/100000)
= 0.00032
Examples
 Find x,
• 2x-3=1
 SOLUTION:
2x-3=20
Bases are same, Equalize the Powers
x-3=0
x=3
Simple Explanation
And Worksheets*
* Answers are not given in questions. But in some cases it is given. The teacher
Has to tell the ANSWERS.
Exponents and Radicals (video)
Exponents and Radicals (Class 8th)

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Exponents and Radicals (Class 8th)

  • 2. Introduction  an = a * a * a * a…* a (where there are n factors)  The number ‘a’ is the base and ‘n’ is the exponent.  For any non-zero rational number a, we define a0 = 1  Let a be any non-zero rational number and n be a positive integer, then we define a-n = 1/an
  • 3. Laws of Exponents aman = am+n When multiplying two powers of the same base, add the exponents. am/ an = am – n When dividing two powers of the same base, subtract the exponents. (am)n = amn When raising a power to a power, multiply the exponents.
  • 4. Laws of Exponents (ab)n = anbn When raising a product to a power, raise each factor to the power. (a/b)n = an / bn When raising a quotient to a power, raise both the numerator and denominator to the power. (a/b)-n = (b/a)n When raising a quotient to a negative power, take the reciprocal and change the power to a positive. a-m / b-n = bm / an To simplify a negative exponent, move it to the opposite position in the fraction. The exponent then becomes positive.
  • 5. nth root If n is any positive integer, then the principal nth root of a is defined as: If n is even, a and b must be positive. means nn a b b a 
  • 6. Properties of nth roots if n is odd | | if n is even n n n n n n m n mn n n n n ab a b a a b b a a a a a a     
  • 7. Rational Exponents  For any rational exponent m/n in lowest terms, where m and n are integers and n>0, we define: If n is even, then we require that a ≥ 0.  am is called the radicand  n is called the index  is called the Radical  In general, am/n=(am)1/n  In general, a-m/n= 1/am/n =1/(am)1/n / nm n m a a / nm n m a a
  • 8. Examples  Express in exponential form: • √7  SOLUTION: √7 = 7 ½  Express in radical form: • (5) 1/3  SOLUTION: (5) 1/3 = 𝟑 𝟓
  • 9. Examples  Find the value of • (125) 2/3  SOLUTION: (125) 2/3 = ( 𝟑 𝟏𝟐𝟓)2 = 52 = 25
  • 10. Examples  Find the value of • (27)-2/3  SOLUTION: (27) -2/3 = ( 𝟑 𝟐𝟕)-2 = 3-2 = 1/9
  • 11. Examples  Find the value of • (6) ½ x (6) 3/2  SOLUTION: (6) ½ + 3/2 = 6 4/2 = 62 = 36
  • 12. Examples  Simplify • (27)6/5 /(27) 1/5  SOLUTION: (27) 6/5- 1/5 =(27)5/5 =27
  • 13. Examples  Evaluate • (0.000064) 5/6  SOLUTION: (64/1000000) 5/6 = [(2/10)6]5/6 =(2/10)5 = (32/100000) = 0.00032
  • 14. Examples  Find x, • 2x-3=1  SOLUTION: 2x-3=20 Bases are same, Equalize the Powers x-3=0 x=3
  • 15. Simple Explanation And Worksheets* * Answers are not given in questions. But in some cases it is given. The teacher Has to tell the ANSWERS.