LAWS OF EXPONENTS

Laws of exponents arise out of certain basic ideas. A positive exponent of a number indicate how many times we use that number in a multiplication whereas a negative exponent suggests us how many times we use that number in a division, since the opposite of multiplying is dividing.

∙ Product law

According to this law, when multiplying two powers that have the same base, we can add the exponents.

a^m ⋅ aⁿ = a^{m + n}

Example :

3⁴ ⋅ 3⁵ = 3^{4 + 5} = 3⁹

∙ Quotient Law

According to this law, when dividing two powers that have the same base we can subtract the exponents.

a^m ÷ aⁿ = a^{m - n}

Example :

3⁹ ÷ 3² = 3^{9 - 7} = 3²

where a (a ≠ 0), m, n are integers. Note that the base should be the same in both the quantities. How does it work? Study the following examples.

∙ Power Law

According to this law, when raising a power to another power, we can just multiply the exponents.

(a^m )ⁿ = a^mn

Example :

(3²)⁵ = 3^{2 x 5} = 3¹⁰

Some other laws of exponents : 

1. A product two or more numbers raised to a power equals the product of each number raised to that power. 

If p and q are any nonzero real numbers and n is any integer, then

(pq)ⁿ = pⁿ ⋅ qⁿ

Example :

(4 ⋅ 7)³ = 4³ ⋅ 7³

2. A quotient raised to a positive power equals the quotient of each base raised to that power.

If p and q are any nonzero real numbers and n is a positive integer, then

(p/q)ⁿ = pⁿ/qⁿ

Example :

(2/3)⁴ = 2⁴/3⁴

3. A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power.

If p and q are any nonzero real numbers and m is a positive integer, then

(p/q)^-m = (q/p)^m

Example :

(5/3)^-2 = (3/5)²

4. If a power is moved from numerator to denominator or denominator to numerator, the sign of the exponent has to be changed.

y^-n = 1/yⁿ

Example :

5^-2 = 1/5²

5. For any nonzero base, if the exponent is zero, its value is 1.

x⁰ = 1

Example :

3⁰ = 1

6. For any base base, if there is no exponent, the exponent is assumed to be 1.

x¹ = x

Example :

3¹ = 3

7. If an exponent is transferred from one side of the equation to the other side of the equation, reciprocal of the exponent has to be taken. 

x^m/n = y ----> x = y^n/m

Example :

x^2/3 = 3

x =  3^3/2

8. If two powers are equal with the same base, exponents can be equated.

a^x = a^y ----> x = y

Example :

3^m = 3⁵ ----> m = 5

9. If two powers are equal with the same exponent, bases can be equated.

x^a = y^a ----> x = y

Example :

k³ = 5³ ----> k = 5

Important Note

(-4)² and -4² are same ?

No, there is a difference between (-4)² and -4².

In (-4)², order of operations (PEMDAS) says to take first.

(-4)² = (-4) ⋅ (-4)

(-4)² = 16

Without parentheses, exponents take precedence :

-4² = -4 ⋅ 4

-4² = -16

Sometimes, the result will be same as in (-2)³ and -2³.

For a negative number with odd exponent, the result is always negative.

Solved Problems

Example 1 :

Find the value of (i) 4^-3 (ii) 1/2^-3 (iii) (-2)⁵ x (-2)^-3 (iv) 3²/3^-2.

Solution :

(i) 4^-3 :

= 1/4³

= 1/(4 x 4 x 4)

= 1/64

(ii) 1/2^-3 :

= 2³

= 2 x 2 x 2

= 8

(iii) (-2)⁵ x (-2)^-3 :

= (-2)^{5 - 3}

= (-2)²

= -2 x -2

= 4

(iv) 3²/3^-2 :

= 3²/3^-2

= 3² x 3²

= 9 x 9

Example 2 :

Simplify and write the answer in exponential form: 

(i) (3⁵ ÷ 3⁸)⁵ x 3^-5 (ii) (-3)⁴ x (5/3)⁴

Solution :

(i) (3⁵ ÷ 3⁸)⁵ x 3^-5 :

= (3^{5 - 8})⁵ x 3^-5

= (3^-3)⁵ x 3^-5

= 3^{-3 x 5} x 3^-5

= 3^-15 x 3^-5

= 3^{-15 - 5}

= = 3^-20

(ii) (-3)⁴ x (5/3)⁴ :

= 3⁴ x 5⁴/3⁴

= 5⁴

Example 3 :

Find x so that (-7)^{x + 2} x (-7)⁵ = (-7)¹⁰.

Solution :

(-7)^{x + 2} x (-7)⁵ = (-7)¹⁰

(-7)^{x + 2 + 5} = (-7)¹⁰

(-7)^{x + 7} = (-7)¹⁰

Since the bases are equal, we can equate the exponents. 

x + 7 = 10

Subtract 7 from each side.

x = 3