PRODUCT RULE OF EXPONENTS

Product Rule of Exponents :

When multiplying exponential expressions that have the same base, add the exponents.

Simplify each of the following.

Example 1 :

a ⋅ a² ⋅ a³

Solution :

Using product rule a^m × aⁿ = a^m+n

= a ⋅ a² ⋅ a³

= a¹⁺² ⋅ a³

= a³ ⋅ a³

= a³⁺³

= a⁶

Example 2 :

(2a²b)(4ab²)

Solution :

= (2a²b)(4ab²)

Multiply the coefficients,

= (2 × 4) (a²b) (ab²)

= 8(a²b) (ab²)

Combining like terms,

= 8(a² ⋅ a) (b ⋅ b²)

By  a^m × aⁿ = a^m+n, we get

= 8(a²⁺¹) (b¹⁺²)

= 8a³b³

Example 3 :

(6x²)(-3x⁵)

Solution :

= (6x²)(-3x⁵)

Multiply the coefficients,

= (6 × (-3)) (x²) (x⁵)

By  a^m × aⁿ = a^m+n, we get

= -18(x²⁺⁵)

= -18x⁷

Example 4 :

b³ ⋅ b⁴ ⋅ b⁷⋅ b

Solution :

= b³ ⋅ b⁴ ⋅ b⁷⋅ b

= b³⁺⁴ ⋅ b⁷⁺¹

= b⁷ ⋅ b⁸

= b⁷⁺⁸

= b¹⁵

Example 5 :

(3x³) (3x⁴) (-3x²)

Solution :

= (3x³) (3x⁴) (-3x²)

Multiply the coefficients,

= (3 × 3 × 3) (x³) (x⁴) (x²)

= 27 (x³⁺⁴) ⋅ (x²)

= 27 (x⁷ ⋅ x²)

= 27x⁷⁺²

= 27x⁹

Use the product rule to rewrite each expression as a single exponent.

Example 6 :

(-5)^-10 ⋅ (-5)¹⁵

Solution :

= (-5)^-10 ⋅ (-5)¹⁵

= (-5)^-10+15

= (-5)⁵

= -3125

Example 7 :

Solution :

Example 8 :

(1.4)^-12 ⋅ (1.4)⁵

Solution :

= (1.4)^-12 ⋅ (1.4)⁵

= (1.4)^-12+5

= (1.4)^-7

= 0.09486

Example 9 :

Solution :

Example 10 :

(-13)⁰ ⋅ (-13)^-19

Solution :

= (-13)⁰ ⋅ (-13)^-19

= (-13)^0-19

= (-13)^-19

= -6.84013

Example 11 :

8^-14 ⋅ 8⁴

Solution :

= 8^-14 ⋅ 8⁴

= 8^-14+4

= 8^-10

= 9.313

Find the value of x :

Example 12 :

10^x ⋅ 10^-9 = 10¹¹

Solution :

10^x ⋅ 10^-9 = 10¹¹

10^x-9 = 10¹¹

Since bases are the same, equate the powers.

x - 9 = 11

x = 11 + 9

x = 20

Example 13 :

Solution :

Example 14 :

(-2.9)^-13 ⋅ (-2.9)^x = (-2.9)^-5

Solution :

(-2.9)^-13 ⋅ (-2.9)^x = (-2.9)^-5

(-2.9)^-13+x = (-2.9)^-5

Since bases are the same, equate the powers.

-13 + x = -5

x = -5 + 13

x = 8