Laws of Exponents

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

aᵐ × aⁿ = am+n${}^{m+n}$

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

(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

1. 5³ ×5⁶

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

= 53+6${}^{3+6}$, [here the exponents are added] 

= (-7)10+12${}^{10+12}$, [Exponents are added] 

=(12$\frac{1}{2}$)⁷

= 32+5${}^{2+5}$

= 3⁷

= (-2)7+3${}^{7+3}$

= (-2)10${}^{10}$

6. (49$\frac{4}{9}$)³ × (49$\frac{4}{9}$)²

= (49$\frac{4}{9}$)⁵

multiplied; the product is obtained by adding the exponent.

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 = 2724$\frac{2^7}{2^4}$

5⁶ ÷ 5² = 5652$\frac{5^6}{5^2}$

a⁵ ÷ a³ = a5a3$\frac{a^5}{a^3}$

aᵐ ÷ aⁿ = aman$\frac{a^m}{a^n}$ = am−n${}^{m-n}$

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

aᵐ ÷ aⁿ = aman$\frac{a^m}{a^n}$ = a−(n−m)${}^{-(n-m)}$

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

                             = 710−8${}^{10-8}$, [here exponents are subtracted] 

2. p⁶ ÷ p¹ = p6p1$\frac{p^6}{p^1}$

3. 4⁴ ÷ 4² = 4442$\frac{4^4}{4^2}$

                = 44−2${}^{4-2}$, [here exponents are subtracted] 

4. 10² ÷ 10⁴ = 102104$\frac{{10}^2}{{10}^4}$

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

= 53−1${}^{3-1}$

= 5²

6. (3)5(3)2$\frac{(3{)}^5}{(3{)}^2}$

= 3³

7. (−5)9(−5)6$\frac{(-5{)}^9}{(-5{)}^6}$

= (-5)³

8. (72$\frac{7}{2}$)⁸ ÷ (72$\frac{7}{2}$)⁵

= (72$\frac{7}{2}$)³

In power of a power you need multiply the powers.

(i) (2³)⁴

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

i.e. (2³)⁴ = 2³ × 2³ × 2³ × 2³

=23+3+3+3${}^{3+3+3+3}$

=2¹²

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

(ii) (2³)²

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

i.e. (2³)² = 2³ × 2³

= 23+3${}^{3+3}$, [since aᵐ × aⁿ = am+n${}^{m+n}$] 

Note: Here, we see that 6 is the product of 3 and 2 i.e,

(iii) (4−2${}^{-2}$)³

= 4−6${}^{-6}$

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

1.(3²)⁴ = 32×4${}^{2\times 4}$ = 3⁸