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.

aᵐ × 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

aᵐ × 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)

= 5\(^{3 + 6}\), [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,

aᵐ ÷ aⁿ = aᵐ/aⁿ = a\(^{m - n}\)

Note 1:

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

aᵐ ÷ aⁿ = aᵐ/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

aᵐ ÷ aⁿ = a\(^{m - n}\) if m < n, then aᵐ ÷ aⁿ = \(\frac{1}{a^{n - m}}\)

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

For example:

1. 7\(^{10}\) ÷ 7⁸ = \(\frac{7^{10}}{7^{8}}\)

= (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 aᵐ × 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 aᵐ × 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. (aᵐ)⁴ = a\(^{m × 4}\) = a⁴ᵐ

5. (2³)⁶ = 2\(^{3 × 6}\) = 2¹⁸

6. (xᵐ)\(^{-n}\) = 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, (aᵐ)ⁿ=a\(^{m × n}\) = aᵐⁿ. 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.

aᵐ × bᵐ

= (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, then a\(^{-m}\) is the reciprocal of aᵐ, i.e., 

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

again, \(\frac{1}{a^{-m}}\) = aᵐ

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\(^{-3}\)

= \(\frac{1}{10^{-3}}\), [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. \(\frac{1}{3^{-4}}\)

= 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: 8\(^{0}\), ( a/b)\(^{0}\), m\(^{0}\)…....

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

a\(^{0}\) = 1

Similarly, (a/b)\(^{0}\) = 1

Consider the following

a\(^{0}\) = 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 aᵐ × 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 aᵐ ÷ aⁿ =a\(^{m - n}\)] 

= 2

= 1

3. 4\(^{0}\) × 3\(^{0}\)

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

= 1

4. aᵐ × a\(^{-m}\)

= a\(^{m - m}\)

= a\(^{0}\)

= 1

5. 5\(^{0}\) = 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\(^{\frac{1}{1}}\) = 2 (it will remain 2). 

2\(^{\frac{1}{2}}\) =√2 (square root of 2). 

2\(^{\frac{1}{3}}\) =∛2 (cube root of 2). 

2\(^{\frac{1}{4}}\) =∜2 (fourth root of 2). 

2\(^{\frac{1}{5}}\) =⁵√2 (fifth root of 2). 

For example:

1. a\(^{\frac{1}{n}}\), [Here a is called the base and 1/n is called the exponent or power] 

= ⁿ√a [nth root of a]

2. 3\(^{\frac{1}{2}}\) = √3 [square root of 3] 

3. 5\(^{\frac{1}{3}}\) = ∛5 [cube root of 5]

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