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, ($\frac{a}{b}$)ᵐ × ($\frac{a}{b}$)ⁿ = ($\frac{a}{b}$)$^{m + n}$

\[(\frac{a}{b})^{m} \times (\frac{a}{b})^{n} = (\frac{a}{b})^{m + n}\]

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

$Multiplying Powers with same Base, Laws of Exponents$

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. $(\frac{1}{2})^{4}$ × $(\frac{1}{2})^{3}$

=[($\frac{1}{2}$) × ($\frac{1}{2}$) × ($\frac{1}{2}$) × ($\frac{1}{2}$)] × [($\frac{1}{2}$) × ($\frac{1}{2}$) × ($\frac{1}{2}$)]

=($\frac{1}{2}$)$^{4 + 3}$

=($\frac{1}{2}$)⁷

4. 3² × 3⁵

= 3$^{2 + 5}$

= 3⁷

5. (-2)⁷ × (-2)³

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

= (-2)$^{10}$

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

= ($\frac{4}{9}$)$^{3 + 2}$

= ($\frac{4}{9}$)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 = $\frac{2^{7}}{2^{4}}$

= $\frac{2 × 2 × 2 × 2 × 2 × 2 × 2}{2 × 2 × 2 × 2}$

= 2$^{7 - 4}$

= 2³

5⁶ ÷ 5² = $\frac{5^{6}}{5^{2}}$

= = $\frac{5 × 5 × 5 × 5 × 5 × 5}{5 × 5}$

= 5$^{6 - 2}$

= 5⁴

10⁵ ÷ 10³ = $\frac{10^{5}}{10^{3}}$

= $\frac{10 × 10 × 10 × 10 × 10}{10 × 10 × 10}$

= 10$^{5 - 3}$

= 10²

7⁴ ÷ 7⁵ = $\frac{7^{4}}{7^{5}}$

= $\frac{7 × 7 × 7 × 7}{7 × 7 × 7 × 7 × 7}$

= 7$^{4 - 5}$

= 7$^{-1}$

Let a be a non zero number, then

a⁵ ÷ a³ = $\frac{a^{5}}{a^{3}}$

= $\frac{a × a × a × a × a}{a × a × a}$

= a$^{5 - 3}$

= a²

again, a³ ÷ a⁵ = $\frac{a^{3}}{a^{5}}$

= $\frac{a × a × a}{a × a × a × a × a}$

= a$^{-(5 - 3)}$

= a$^{-2}$

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

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

Note 1:

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

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

$Dividing Powers with the same Base, Laws of Exponents$

For example:

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

= $\frac{7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7}{7 × 7 × 7 × 7 × 7 × 7 × 7 × 7}$

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

= 7²

2. p⁶ ÷ p¹ = $\frac{p^{6}}{p^{1}}$

= $\frac{p × p × p × p × p × p}{p}$

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

= p⁵

3. 4⁴ ÷ 4² = $\frac{4^{4}}{4^{2}}$

= $\frac{4 × 4 × 4 × 4}{4 × 4}$

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

= 4²

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

= $\frac{10 × 10}{10 × 10 × 10 × 10}$

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

= 10$^{-2}$

5. 5³ ÷ 5¹

= 5$^{3 - 1}$

= 5²

6. $\frac{(3)^{5}}{(3)^{2}}$

= 3$^{5 - 2}$

= 3³

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

= (-5)$^{9 - 6}$

= (-5)³

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

= ($\frac{7}{2}$)$^{8 - 5}$

= ($\frac{7}{2}$)³

$Laws of Exponents or Indices$

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$^{m × n}$ = 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 {($\frac{a}{b}$)ᵐ}ⁿ = ($\frac{a}{b}$)$^{mn}$

$Power of a Power, Laws of Exponents$

For example:

[($\frac{-2}{5}$)³]²

= ($\frac{-2}{5}$)$^{3 × 2}$

= ($\frac{-2}{5}$)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.

$Multiplying Powers with the same Exponents, Exponent Rules$

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]

aᵐ × bᵐ = (ab)ᵐ

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 $\frac{p}{q}$ then ($\frac{p}{q}$)$^{-m}$ = $\frac{1}{(\frac{p}{q})^{m}}$ = ($\frac{q}{p}$)ᵐ

again, $\frac{1}{a^{-m}}$ = aᵐ

Similarly, ($\frac{a}{b}$)$^{-n}$ = ($\frac{b}{a}$)ⁿ, where n is a positive integer

Consider the following

2$^{-1}$ = $\frac{1}{2}$

2$^{-2}$ = $\frac{1}{2^{2}}$ = $\frac{1}{2}$ × $\frac{1}{2}$ = $\frac{1}{4}$

2$^{-3}$ = $\frac{1}{2^{3}}$ = $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ = $\frac{1}{8}$

2$^{-4}$ = $\frac{1}{2^{4}}$ = $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ = $\frac{1}{16}$

2$^{-5}$ = $\frac{1}{2^{5}}$ = $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ × $\frac{1}{2}$ = $\frac{1}{32}$

[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]

$Negative Exponents, Laws of Exponents$

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]

= $\frac{1}{10}$ × $\frac{1}{10}$ × $\frac{1}{10}$, [Here 10 is multiplied to itself 3 times]

= $\frac{1}{1000}$

2. (-2)$^{-4}$

= $\frac{1}{(-2)^{4}}$ [Here we can see that 1 is in the numerator and in the denominator (-2)⁴]

= (- $\frac{1}{2}$) × (- $\frac{1}{2}$) × (- $\frac{1}{2}$) × (- $\frac{1}{2}$)

= $\frac{1}{16}$

3. 2$^{-5}$

= $\frac{1}{2^{5}}$

= $\frac{1}{2}$ × $\frac{1}{2}$

= $\frac{1}{4}$

4. $\frac{1}{3^{-4}}$

= 3⁴

= 3 × 3 × 3 × 3

= 81

5. (-7)$^{-3}$

= $\frac{1}{(-7)^{3}}$

6. ($\frac{3}{5}$)$^{-3}$

= ($\frac{5}{3}$)³

7. (-$\frac{7}{2}$)$^{-2}$

= (-$\frac{2}{7}$)²

6. Power with Exponent Zero

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

For example: 8$^{0}$, ($\frac{a}{b}$)$^{0}$, m$^{0}$…....

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

a$^{0}$ = 1

Similarly, ($\frac{a}{b}$)$^{0}$ = 1

Consider the following

a$^{0}$ = 1 [anything to the power 0 is 1]

($\frac{a}{b}$)$^{0}$ = 1

($\frac{-2}{3}$)$^{0}$ = 1

(-3)$^{0}$ = 1

$Power with Exponent Zero, Laws of Exponents$

For example:

1. ($\frac{2}{3}$)³ × ($\frac{2}{3}$)$^{-3}$

= ($\frac{2}{3}$)$^{3 + (-3)}$, [Here we know that aᵐ × aⁿ = a$^{m + n}$]

= ($\frac{2}{3}$)$^{3 - 3}$

= ($\frac{2}{3}$)$^{0}$

= 1

2. 2⁵ ÷ 2⁵

= $\frac{2^{5}}{2^{5}}$

= $\frac{2 × 2 × 2 × 2 × 2}{2 × 2 × 2 × 2 × 2}$

= 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. ($\frac{-4}{9}$)$^{0}$ = 1

7. (-41)$^{0}$ = 1

8. ($\frac{3}{7}$)$^{0}$ = 1

7. Fractional Exponent

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

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

= $\sqrt[n]{a}$, [nth root of a]

\[a^{\frac{1}{n}} = \sqrt[n]{a}\]

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$^{\frac{1}{5}}$ = $\sqrt[5]{2}$ (fifth root of 2).

$Fractional Exponent, Laws of Exponents$

For example:

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

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

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

4. 10$^{\frac{1}{3}}$ = ∛10 [cube root of 10]

5. 21$^{\frac{1}{7}}$ = $\sqrt[7]{21}$ [Seventh root of 21]

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