# LAWS OF EXPONENTS

## Laws of Exponent

Instead of writing the repeating the same, we can use the exponents

## Example Problems on Laws of Exponents

Simplify each of the following.

Problem 1 :

⋅ a2 ⋅ a3

Solution :

= a ⋅ a2 ⋅ a3

Since the base is same for all three terms, we use only one base and add the powers.

= a(1+2+3)

= a6

Problem 2 :

(2 a2 b) ⋅ (4ab2)

Solution :

= (2 a2 b) ⋅ (4ab2)

Multiplying the coefficients = 2 x 4 ==> 8

Multiplying a related terms = a2⋅ a = a3

Multiplying b related terms = b ⋅ b2 = b3

Since the base is same for all three terms, we use only one base and add the powers.

= 8a3b3

Problem 3 :

(6x2) ⋅ (-3x5)

Solution :

= (6x2) ⋅ (-3x5)

= -18 x2+5

= -18x7

Problem 3 :

(6x2) ⋅ (-3x5)

Solution :

= (6x2) ⋅ (-3x5)

= -18 x2+5

= -18x7

Problem 4 :

(2x2 y3)2

Solution :

= (2x2 y3)2

Distributing the power, we get

22 (x2) (y3)2

= 22 x4 y6

= 4x4 y6

Problem 5 :

Solution :

## Rules with Negative Exponent

(-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.

## Problems with Negative Exponents

Problem 6 :

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

Problem 7 :

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

Problem 8 :

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

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