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⋅ an = am + n

Example :

34 ⋅ 35 = 34 + 5 = 39

∙ Quotient Law

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

a÷ an = am - n

Example :

39 ÷ 32 = 39 - 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)n = amn

Example :

(32)5 = 32 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. 

xm/n = y ----> x = yn/m

Example :

x2/3 = 3

x =  33/2

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

ax = ay ----> x = y

Example :

3m = 35 ----> m = 5

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

xa = ya ----> 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)= (-4) ⋅ (-4)

(-4)= 16

Without parentheses, exponents take precedence :

-4= -4 ⋅ 4

-4= -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) (3÷ 38)5 x 3-5 (ii) (-3)x (5/3)4

Solution :

(i) (3÷ 38)5 x 3-5 :

= (35 - 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)x (5/3)4 :

= 3x 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

Recent Articles

  1. Solving Equations with Rational Coefficients Worksheet

    Jun 05, 23 09:20 PM

    Solving Equations with Rational Coefficients Worksheet

    Read More

  2. GL Assessment 11 Plus Maths Practice Papers

    Jun 05, 23 08:05 AM

    GL Assessment 11 Plus Maths Practice Papers

    Read More

  3. GL Assessment 11 Plus Practice Papers Maths

    Jun 05, 23 07:42 AM

    GL Assessment 11 Plus Practice Papers Maths

    Read More