The laws of exponents are explained here along with their examples.

Laws of Exponents

The laws of exponents are explained here along with their examples.

1. Multiplying powers with same base

For example: x² × x³, 2³ × 2⁵, (-3)² × (-3)⁴

In multiplication of exponents if the bases are same then we need to add the exponents.

Consider the following:

1. 2³ × 2²= (2 × 2 × 2) × (2 × 2) = 2^{3 + 2} = 2⁵

2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 3^{4 + 2} = 3⁶

3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]

= (-3)^{3 + 4} = (-3)⁷

4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m) = m^{5 + 3} = m⁸

From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.

a^m × aⁿ = a^m+n

In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

a^m × aⁿ = a^{m + n}

Similarly, (a/b)^m × (a/b)ⁿ = (a/b)^{m + n}

Note:

(i) Exponents can be added only when the bases are same.

(ii) Exponents cannot be added if the bases are not same like

m⁵ × n⁷, 2³ × 3⁴

For example:

1. 5³ ×5⁶

= (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5)

= 5³⁺⁶[here the exponents are added]

= 5⁹

2. (-7)¹⁰ × (-7)¹²

= [(-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)] × [( -7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)].

= (-7)¹⁰⁺¹² [exponents are added]

= (-7)²²

3. (1/2)⁴× ( 1/2)³

=[(1/2) × ( 1/2) × ( 1/2) × ( 1/2)] × [ ( 1/2) × ( 1/2) × ( 1/2)]

=(1/2)^{4 + 3}

=(1/2)⁷

4. 3² × 3⁵

= 3^{2 + 5}

= 3⁷

5. (-2)⁷ × (-2)³

= (-2)^{7 + 3}

= (-2)¹⁰

6. (4/9)³ × (4/9)²

= (4/9)^{3 + 2}

= (4/9)⁵

We observe that the two numbers with the same base are

multiplied; the product is obtained by adding the exponent.

2. Dividing powers with the same base

For example:

3⁵ ÷ 3¹, 2² ÷ 2¹, 5(²) ÷ 5³

In division if the bases are same then we need to subtract the exponents.

Consider the following:

2⁷ ÷ 2⁴ = 2⁷/2⁴ = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 2^{7 - 4}

= 2³

5⁶ ÷ 5² = 5⁶/5² = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 5^6-2 = 5⁴

10⁵ ÷ 10³ = 10⁵/10³ = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 10^{5 - 3}

= 10²

7⁴ ÷ 7⁵ = 7⁴/7⁵ = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 7^{4 - 5} = 7^-1

Let a be a non zero number, then

a⁵ ÷ a³ = a⁵/a³ = (a × a × a × a × a)/(a × a × a) = a^{5 - 3} = a²

again, a³ ÷ a⁵ = a³/a⁵ = (a × a × a)/(a × a × a × a × a) = a^{-(5 - 3 )} = a^-2

Thus, in general, for any non-zero integer a,

a^m ÷ aⁿ = a^m/aⁿ = a^{m - n}

Note 1:

Where m and n are whole numbers and m > n;

a^m ÷ aⁿ = a^m/aⁿ = a^{-(n - m)}

Note 2:

Where m and n are whole numbers and m < n;

We can generalize that if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m > n, then

a^m ÷ aⁿ = a^{m - n} if m < n, then a^m ÷ aⁿ = 1/a^{n - m}

Similarly, (a/b)^m ÷ (a/b)ⁿ = (a/b)^{m -n}

For example:

1. 7¹⁰ ÷ 7⁸ = 7¹⁰/7⁸

= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)

= 7^{10 - 8} [here exponents are subtracted]

= 7²

2. p⁶ ÷ p¹=p⁶/p¹

= (p × p × p × p × p × p)/p

= p^{6 - 1} [here exponents are subtracted]

= p⁵

3. 4⁴ ÷ 4² = 4⁴/4²

= (4 × 4 × 4 × 4)/(4 × 4)

= 4^{4 - 2} [here exponents are subtracted]

= 4²

4. 10² ÷ 10⁴ = 10²/10⁴

= (10 × 10)/(10 × 10 × 10 × 10)

= 10^{-(4 - 2)} [See note (2)]

= 10^-2

5. 5³ ÷ 5¹

= 5^{3 - 1}

= 5²

6. 3⁵/3²

= 3^{5 - 2}

= 3³

7. (-5)⁹/(-5)⁶

= (-5)^{9 - 6}

= (-5)³

8. (7/2)⁸ ÷ (7/2)⁵

= (7/2)^{8 - 5}

= (7/2)³

3. Power of a power

For example: (2³)², (5²)⁶, (3² )^-3

In power of a power you need multiply the powers.

Consider the following

(i) (2³)⁴

Now, (2³)⁴ means 2³ is multiplied four times

i.e. (2³)⁴ = 2³ × 2³ × 2³ × 2³

=2^{3 + 3 + 3 + 3}

=2¹²

Note: by law (l), since a^m × aⁿ = a^{m + n}.

(ii) (2³)²

Similarly, now (2³)² means 2³ is multiplied two times

i.e. (2³)² = 2³ × 2³

= 2^{3 + 3} [since a^m × aⁿ = a^{m + n}]

= 2⁶

Note: Here, we see that 6 is the product of 3 and 2 i.e,(2³)² = 2^{3 × 2} = 2⁶

(iii)(4^-2)³

Similarly, now (4^-2 )³ means 4^-2 is multiplied three times

i.e. (4^-2)³ =4^-2 × 4^-2 × 4^-2

=4^{-2 + (-2) + (-2)}

=4^{-2 -2 -2}

=4^-6

Note: Here, we see that -6 is the product of -2 and 3 i.e, (4^-2)³

= 4^{-2 × 3} =4^-6

For example:

1.(3²)⁴ = 3^{2 × 4} = 3⁸

2. (5³)⁶ = 5^{3 × 6} = 5¹⁸

3. (4³)⁸ = 4^{3 × 8} = 4²⁴

4. (a^m)⁴ = a^{m × 4} = a^4m

5. (2³)⁶ = 2^{3 × 6} = 2¹⁸

6. (x^m)^-n = x^{m × (-n)} = x^-mn

7. (5²)⁷ = 5^{2 × 7} = 5¹⁴

8. [(-3)⁴]² = (-3)^{4 × 2} = (-3)⁸

In general, for any non-integer a, (a^m)ⁿ=a^{m × n} =a^mn.Thus where m and n are whole numbers.

If ‘a’ is a non-zero rational number and m and n are positive integers, then {(a/b)^m}ⁿ = (a/b) ^mn

For example:

[(-2/5)³]²

= (-2/5)^{3 × 2}

= (-2/5)⁶

4. Multiplying powers with the same exponents

For example: 3² × 2², 5³ × 7³

We consider the product of 4² and 3², which have different bases, but the same exponents.

(i) 4² × 3² [here the powers are same and the bases are different]

= (4 × 4)×(3 × 3)

= (4 × 3)×(4 × 3)

= 12 × 12

= 12²

Here, we observe that in 12², the base is the product of bases 4 and 3.

We consider,

(ii) 4³ × 2³

=(4 × 4 × 4)×(2 × 2 × 2)

=(4 × 2)× ( 4 × 2)× (4 × 2)

=8 × 8 × 8

=8³

(iii) We also have, 2³ × a³

= (2× 2 × 2)×(a × a × a)

= (2 × a)×(2 × a)×(2 × a)

= (2 × a)³

= (2a)³ [here 2 × a = 2a]

(iv) Similarly, we have, a³ × b³

= (a × a × a)×(b × b × b)

= (a × b)× (a × b)× (a × b)

= (a × b)³

= (ab)³ [here a × b = ab]

Note: In general, for any non-zero integer a, b.

a^m × b^m

= (a × b)^m

= (ab)^m [here a × b = ab]

Note: Where m is any whole number.

(-a)³ × (-b)³

= [(-a) × (-a) × (-a)]×[(-b) × (-b) × (-b)]

= [(-a) × (-b)]× [(-a) × (-b)]× [(-a) × (-b)]

= [(-a)×(-b)]³

= (ab)³ [here a × b = ab and two negative become positive,

(-) × (-) = +]

5. Negative Exponents

If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.

If ‘a’ is a non-zero integer or a non-zero rational number and m is a positive integers, thena^-m is the reciprocal of a^m, i.e.,

a^-m = 1/a^m, if we take ‘a’ as p/q then (p/q)^-m = 1/(p/q)^m = (q/p)^m

again, 1/a^-m = a^m

Similarly, (a/b)^-n = (b/a)ⁿ, where n is a positive integer

Consider the following

2^-1 = 1/2

2^-2 = 1/2² = 1/2 × 1/2 = 1/4

2^-3 = 1/2³ = 1/2 × 1/2 × 1/2 = 1/8

2^-4 = 1/2⁴ = 1/2 × 1/2 × 1/2 × 1/2 = 1/16

2^-5 = 1/2⁵ = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32

[So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 2^-5. In other words negative exponent is the reciprocal of positive exponent]

For example:

1. 10^-3

= 1/10^-3 [here we can see that 1 is in the numerator and in the denominator 10³ as we know that negative exponent is the reciprocal]

= 1/10 × 1/10 × 1/10 [here 10 is multiplied to itself 3 times]

= 1/1000

2. (-2)^-4

= 1/(-2)⁴ [here we can see that 1 is in the numerator and in the denominator (-2)⁴]

=(- 1/2) × (- 1/2) × (- 1/2) × (- 1/2)

= 1/16

3. 2^-5

= 1/2⁵

= 1/2 × 1/2

= 1/4

4. 1/3^-4

= 3⁴

= 3 × 3 × 3 × 3

= 81

5. (-7)^-3

= 1/(-7)³

6. (3/5^-3

= (5/3)³

7. (-7/2)^-2

= (-2/7)²

6. Power with exponent zero

If the exponent is 0 then you get the result 1 whatever the base is.

For example: 8⁰, ( a/b)⁰, m⁰….

If ‘a’ is a non-zero integer or a non-zero rational number then,

a⁰ = 1

Similarly, (a/b)⁰ = 1

Consider the following

a⁰ = 1[anything to the power 0 is 1]

(a/b)⁰ = 1

(-2/3)⁰ = 1

(-3)⁰ = 1

For example:

1. (2/3)³ × (2/3)^-3

= (2/3)³ + (-3) [here we know that a^m × aⁿ = a^{m+ n}]

= (2/3)^{3 - 3}

= (2/3)⁰

= 1

2. 2⁵ ÷ 2⁵

= 2⁵/2⁵

= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2)

= 2^{5 - 5} [here by the law a^m ÷ aⁿ =a^{m - n}]

= 2⁰

= 1

3. 4⁰ × 3⁰

= 1 × 1[here as we know anything to the power 0 is 1]

= 1

4. a^m × a^-m

= a^{m - m}

= a⁰

= 1

5. 5⁰ = 1

6. (-4/9)⁰ = 1

7. (-41)⁰ = 1

8. (3/7)⁰ = 1

7. Fractional Exponent

In fractional exponent we observe that the exponent is in fraction form.

Consider the following:

2^1/1 = 2 (it will remain 2).

2^1/2 =√2 (square root of 2).

2^1/3 =∛2 (cube root of 2).

2^1/4 =∜2 (fourth root of 2).

2^1/5 =⁵√2 (fifth root of 2).

For example:

1. a^1/n [Here a is called the base and 1/n is called the exponent or power]

= ⁿ√a [nth root of a]

2. 3^1/2= √3 [square root of 3]

3. 5^1/3 = ∛5 [cube root of 5]

Exponents

Exponents
Laws of Exponents
Rational Exponent
Integral Exponents of a Rational Numbers
Solved Examples on Exponents
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Consider the following: 1. 2³ × 2²= (2 × 2 × 2) × (2 × 2) = 2^{3 + 2} = 2⁵ 2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 3^{4 + 2} = 3⁶ 3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)] = (-3)^{3 + 4} = (-3)⁷ 4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m) = m^{5 + 3} = m⁸ From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added. a^m × aⁿ = a^m+n In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then a^m × aⁿ = a^{m + n} Similarly, (a/b)^m × (a/b)ⁿ = (a/b)^{m + n} Note: (i) Exponents can be added only when the bases are same. (ii) Exponents cannot be added if the bases are not same like m⁵ × n⁷, 2³ × 3⁴ For example: 1. 5³ ×5⁶ = (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5) = 5³⁺⁶[here the exponents are added] = 5⁹ 2. (-7)¹⁰ × (-7)¹² = [(-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)] × [( -7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7) × (-7)]. = (-7)¹⁰⁺¹² [exponents are added] = (-7)²² 3. (1/2)⁴× ( 1/2)³ =[(1/2) × ( 1/2) × ( 1/2) × ( 1/2)] × [ ( 1/2) × ( 1/2) × ( 1/2)] =(1/2)^{4 + 3} =(1/2)⁷ 4. 3² × 3⁵ = 3^{2 + 5} = 3⁷ 5. (-2)⁷ × (-2)³ = (-2)^{7 + 3} = (-2)¹⁰ 6. (4/9)³ × (4/9)² = (4/9)^{3 + 2} = (4/9)⁵ We observe that the two numbers with the same base are multiplied; the product is obtained by adding the exponent. 2. Dividing powers with the same base For example: 3⁵ ÷ 3¹, 2² ÷ 2¹, 5(²) ÷ 5³ In division if the bases are same then we need to subtract the exponents. Consider the following: 2⁷ ÷ 2⁴ = 2⁷/2⁴ = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 2^{7 - 4} = 2³ 5⁶ ÷ 5² = 5⁶/5² = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 5^6-2 = 5⁴ 10⁵ ÷ 10³ = 10⁵/10³ = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 10^{5 - 3} = 10² 7⁴ ÷ 7⁵ = 7⁴/7⁵ = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 7^{4 - 5} = 7^-1 Let a be a non zero number, then a⁵ ÷ a³ = a⁵/a³ = (a × a × a × a × a)/(a × a × a) = a^{5 - 3} = a² again, a³ ÷ a⁵ = a³/a⁵ = (a × a × a)/(a × a × a × a × a) = a^{-(5 - 3 )} = a^-2 Thus, in general, for any non-zero integer a, a^m ÷ aⁿ = a^m/aⁿ = a^{m - n} Note 1: Where m and n are whole numbers and m > n; a^m ÷ aⁿ = a^m/aⁿ = a^{-(n - m)} Note 2: Where m and n are whole numbers and m < n; We can generalize that if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m > n, then a^m ÷ aⁿ = a^{m - n} if m < n, then a^m ÷ aⁿ = 1/a^{n - m} Similarly, (a/b)^m ÷ (a/b)ⁿ = (a/b)^{m -n} For example: 1. 7¹⁰ ÷ 7⁸ = 7¹⁰/7⁸ = (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7 × 7 × 7 × 7) = 7^{10 - 8} [here exponents are subtracted] = 7² 2. p⁶ ÷ p¹=p⁶/p¹ = (p × p × p × p × p × p)/p = p^{6 - 1} [here exponents are subtracted] = p⁵ 3. 4⁴ ÷ 4² = 4⁴/4² = (4 × 4 × 4 × 4)/(4 × 4) = 4^{4 - 2} [here exponents are subtracted] = 4² 4. 10² ÷ 10⁴ = 10²/10⁴ = (10 × 10)/(10 × 10 × 10 × 10) = 10^{-(4 - 2)} [See note (2)] = 10^-2 5. 5³ ÷ 5¹ = 5^{3 - 1} = 5² 6. 3⁵/3² = 3^{5 - 2} = 3³ 7. (-5)⁹/(-5)⁶ = (-5)^{9 - 6} = (-5)³ 8. (7/2)⁸ ÷ (7/2)⁵ = (7/2)^{8 - 5} = (7/2)³ 3. Power of a power For example: (2³)², (5²)⁶, (3² )^-3 In power of a power you need multiply the powers. Consider the following (i) (2³)⁴ Now, (2³)⁴ means 2³ is multiplied four times i.e. (2³)⁴ = 2³ × 2³ × 2³ × 2³ =2^{3 + 3 + 3 + 3} =2¹² Note: by law (l), since a^m × aⁿ = a^{m + n}. (ii) (2³)² Similarly, now (2³)² means 2³ is multiplied two times i.e. (2³)² = 2³ × 2³ = 2^{3 + 3} [since a^m × aⁿ = a^{m + n}] = 2⁶ Note: Here, we see that 6 is the product of 3 and 2 i.e,(2³)² = 2^{3 × 2} = 2⁶ (iii)(4^-2)³ Similarly, now (4^-2 )³ means 4^-2 is multiplied three times i.e. (4^-2)³ =4^-2 × 4^-2 × 4^-2 =4^{-2 + (-2) + (-2)} =4^{-2 -2 -2} =4^-6 Note: Here, we see that -6 is the product of -2 and 3 i.e, (4^-2)³ = 4^{-2 × 3} =4^-6 For example: 1.(3²)⁴ = 3^{2 × 4} = 3⁸ 2. (5³)⁶ = 5^{3 × 6} = 5¹⁸ 3. (4³)⁸ = 4^{3 × 8} = 4²⁴ 4. (a^m)⁴ = a^{m × 4} = a^4m 5. (2³)⁶ = 2^{3 × 6} = 2¹⁸ 6. (x^m)^-n = x^{m × (-n)} = x^-mn 7. (5²)⁷ = 5^{2 × 7} = 5¹⁴ 8. [(-3)⁴]² = (-3)^{4 × 2} = (-3)⁸ In general, for any non-integer a, (a^m)ⁿ=a^{m × n} =a^mn.Thus where m and n are whole numbers. If ‘a’ is a non-zero rational number and m and n are positive integers, then {(a/b)^m}ⁿ = (a/b) ^mn For example: [(-2/5)³]² = (-2/5)^{3 × 2} = (-2/5)⁶ 4. Multiplying powers with the same exponents For example: 3² × 2², 5³ × 7³ We consider the product of 4² and 3², which have different bases, but the same exponents. (i) 4² × 3² [here the powers are same and the bases are different] = (4 × 4)×(3 × 3) = (4 × 3)×(4 × 3) = 12 × 12 = 12² Here, we observe that in 12², the base is the product of bases 4 and 3. We consider, (ii) 4³ × 2³ =(4 × 4 × 4)×(2 × 2 × 2) =(4 × 2)× ( 4 × 2)× (4 × 2) =8 × 8 × 8 =8³ (iii) We also have, 2³ × a³ = (2× 2 × 2)×(a × a × a) = (2 × a)×(2 × a)×(2 × a) = (2 × a)³ = (2a)³ [here 2 × a = 2a] (iv) Similarly, we have, a³ × b³ = (a × a × a)×(b × b × b) = (a × b)× (a × b)× (a × b) = (a × b)³ = (ab)³ [here a × b = ab] Note: In general, for any non-zero integer a, b. a^m × b^m = (a × b)^m = (ab)^m [here a × b = ab] Note: Where m is any whole number. (-a)³ × (-b)³ = [(-a) × (-a) × (-a)]×[(-b) × (-b) × (-b)] = [(-a) × (-b)]× [(-a) × (-b)]× [(-a) × (-b)] = [(-a)×(-b)]³ = (ab)³ [here a × b = ab and two negative become positive, (-) × (-) = +] 5. Negative Exponents If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator. If ‘a’ is a non-zero integer or a non-zero rational number and m is a positive integers, thena^-m is the reciprocal of a^m, i.e., a^-m = 1/a^m, if we take ‘a’ as p/q then (p/q)^-m = 1/(p/q)^m = (q/p)^m again, 1/a^-m = a^m Similarly, (a/b)^-n = (b/a)ⁿ, where n is a positive integer Consider the following 2^-1 = 1/2 2^-2 = 1/2² = 1/2 × 1/2 = 1/4 2^-3 = 1/2³ = 1/2 × 1/2 × 1/2 = 1/8 2^-4 = 1/2⁴ = 1/2 × 1/2 × 1/2 × 1/2 = 1/16 2^-5 = 1/2⁵ = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32 [So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 2^-5. In other words negative exponent is the reciprocal of positive exponent] For example: 1. 10^-3 = 1/10^-3 [here we can see that 1 is in the numerator and in the denominator 10³ as we know that negative exponent is the reciprocal] = 1/10 × 1/10 × 1/10 [here 10 is multiplied to itself 3 times] = 1/1000 2. (-2)^-4 = 1/(-2)⁴ [here we can see that 1 is in the numerator and in the denominator (-2)⁴] =(- 1/2) × (- 1/2) × (- 1/2) × (- 1/2) = 1/16 3. 2^-5 = 1/2⁵ = 1/2 × 1/2 = 1/4 4. 1/3^-4 = 3⁴ = 3 × 3 × 3 × 3 = 81 5. (-7)^-3 = 1/(-7)³ 6. (3/5^-3 = (5/3)³ 7. (-7/2)^-2 = (-2/7)² 6. Power with exponent zero If the exponent is 0 then you get the result 1 whatever the base is. For example: 8⁰, ( a/b)⁰, m⁰…. If ‘a’ is a non-zero integer or a non-zero rational number then, a⁰ = 1 Similarly, (a/b)⁰ = 1 Consider the following a⁰ = 1[anything to the power 0 is 1] (a/b)⁰ = 1 (-2/3)⁰ = 1 (-3)⁰ = 1 For example: 1. (2/3)³ × (2/3)^-3 = (2/3)³ + (-3) [here we know that a^m × aⁿ = a^{m+ n}] = (2/3)^{3 - 3} = (2/3)⁰ = 1 2. 2⁵ ÷ 2⁵ = 2⁵/2⁵ = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = 2^{5 - 5} [here by the law a^m ÷ aⁿ =a^{m - n}] = 2⁰ = 1 3. 4⁰ × 3⁰ = 1 × 1[here as we know anything to the power 0 is 1] = 1 4. a^m × a^-m = a^{m - m} = a⁰ = 1 5. 5⁰ = 1 6. (-4/9)⁰ = 1 7. (-41)⁰ = 1 8. (3/7)⁰ = 1 7. Fractional Exponent In fractional exponent we observe that the exponent is in fraction form. Consider the following: 2^1/1 = 2 (it will remain 2). 2^1/2 =√2 (square root of 2). 2^1/3 =∛2 (cube root of 2). 2^1/4 =∜2 (fourth root of 2). 2^1/5 =⁵√2 (fifth root of 2). For example: 1. a^1/n [Here a is called the base and 1/n is called the exponent or power] = ⁿ√a [nth root of a] 2. 3^1/2= √3 [square root of 3] 3. 5^1/3 = ∛5 [cube root of 5] Exponents Exponents Laws of Exponents Rational Exponent Integral Exponents of a Rational Numbers Solved Examples on Exponents Practice Test on Exponents Exponents - Worksheets Worksheet on Exponents 8th Grade Math Practice From Laws of Exponents to HOME PAGE © and ™ math-only-math.com. All Rights Reserved. 2010 - 2012.