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

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

(\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 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2}

= 2

^{7 - 4}

= 2³

5⁶ ÷ 5² = $\frac{5^{6}}{5^{2}}$

= =

\frac{5 \times 5 \times 5 \times 5 \times 5 \times 5}{5 \times 5}

= 5

^{6 - 2}

= 5⁴

10⁵ ÷ 10³ =

\frac{10^{5}}{10^{3}}

\frac{10 \times 10 \times 10 \times 10 \times 10}{10 \times 10 \times 10}

= 10

^{5 - 3}

= 10²

7⁴ ÷ 7⁵ =

\frac{7^{4}}{7^{5}}

\frac{7 \times 7 \times 7 \times 7}{7 \times 7 \times 7 \times 7 \times 7}

= 7

^{4 - 5}

= 7

^{- 1}

Let a be a non zero number, then

a⁵ ÷ a³ = $\frac{a^{5}}{a^{3}}$

\frac{a \times a \times a \times a \times a}{a \times a \times a}

= a

^{5 - 3}

= a²

again, a³ ÷ a⁵ =

\frac{a^{3}}{a^{5}}

\frac{a \times a \times a}{a \times a \times a \times a \times 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}

For example:

1. 7

^{10}

÷ 7⁸ =

\frac{7^{10}}{7^{8}}

\frac{7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7}{7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7}

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

= 7²

2. p⁶ ÷ p¹ = $\frac{p^{6}}{p^{1}}$

\frac{p \times p \times p \times p \times p \times p}{p}

= p

^{6 - 1}

, [here exponents are subtracted]

= p⁵

3. 4⁴ ÷ 4² = $\frac{4^{4}}{4^{2}}$

\frac{4 \times 4 \times 4 \times 4}{4 \times 4}

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

= 4²

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

\frac{10 \times 10}{10 \times 10 \times 10 \times 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

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 \times 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,

^{- 2}

)³ = 4

^{- 2 \times 3}

= 4

^{- 6}

For example:

1.(3²)⁴ = 3 $^{2 \times 4}$ = 3⁸

2. (5³)⁶ = 5

^{3 \times 6}

= 5¹⁸

3. (4³)⁸ = 4

^{3 \times 8}

= 4²⁴

4. (aᵐ)⁴ = a

^{m \times 4}

= a⁴ᵐ

5. (2³)⁶ = 2

^{3 \times 6}

= 2¹⁸

6. (xᵐ)

^{- n}

= x

^{m \times - (n)}

= x

^{- m n}

7. (5²)⁷ = 5

^{2 \times 7}

= 5¹⁴

8. [(-3)⁴]² = (-3)

^{4 \times 2}

= (-3)⁸

In general, for any non-integer a, (aᵐ)ⁿ= a

^{m \times n}

= a

^{m n}

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}

)

^{m n}

For example:

[(

\frac{- 2}{5}

)³]²

= (

\frac{- 2}{5}

)

^{3 \times 2}

= (

\frac{- 2}{5}

)⁶

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Laws of Exponents

1. Multiplying Powers with same Base

2. Dividing Powers with the same Base

3. Power of a Power

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1. Multiplying Powers with same Base

2. Dividing Powers with the same Base

3. Power of a Power

About The Author

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