LAWS OF EXPONENTS

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

Simplify each of the following.

Problem 1 :

a ⋅ 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)²  (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)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.

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) (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

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