EVALUATING POWERS

The following properties of exponents can be used to evaluate expressions with powers. 

x^m ⋅ xⁿ  =  x^m+n

x^m ÷ xⁿ  =  x^m-n

(x^m)ⁿ  =  x^mn

(xy)^m  =  x^m ⋅ y^m

(x / y)^m  =  x^m / y^m

x^-m  =  1 / x^m

x^m/n  =  y -----> x  =  y^n/m

(x / y)^-m  =  (y / x)^m

Example 1 :

Evaluate : 

4³ ⋅ 5³

Solution :

=  4³ ⋅ 5³

=  4 ⋅ 4 ⋅ 4 ⋅ 5 ⋅ 5 ⋅ 5

=  64 ⋅ 125

=  8000

Example 2 :

Evaluate :

10⁹ ÷ 10⁶

Solution :

=  10⁹ ÷ 10⁶

=   10^{9 - 6}

=   10³

=  10 ⋅ 10 ⋅ 10

=  1000

Example 3 :

Evaluate :

(3/2)⁵

Solution :

=  (3/2)⁵

=  3⁵ / 2⁵ 

  =  (3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) / (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

  =  243/32

Example 4 :

Evaluate : 

3⁴ / 3^-3

Solution :

=  3⁴ / 3^-3

=  3^{4 - (-3)}

=  3^{4 + 3}

=  3⁷

=  3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3

=  2187

Example 5 :

Evaluate the following expression when p = 2.

(4p)³ ⋅ (2p)² ⋅ p

Solution :

=  (4p)³ ⋅ (2p)² ⋅ p

=  (4³p³) ⋅ (2²p²) ⋅ p

=  (4 ⋅ 4 ⋅ 4 ⋅ p³) ⋅ (2 ⋅ 2 ⋅ p²) ⋅ p

=  (64p³) ⋅ (4p²) ⋅ p

=  (64 ⋅ 4) ⋅ (p³ ⋅ p² ⋅ p)

=  256 ⋅ p^{3 + 2 + 1}

=  256 ⋅ p⁶

Substitute 2 for p. 

=  256 ⋅ 2⁶

=  256 ⋅ (2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

=  256 ⋅ 64

=  16384

Example 6 :

Evaluate : 

[(-2/3)^-2]^-2

Solution :

=  [(-2/3)^-2]^-2

=  (-2/3)^(-2)(-2)

=  (-2/3)⁴

=  (-2)⁴ / 3⁴

=  (-2 ⋅  -2 ⋅  -2 ⋅ -2) / (3 ⋅ 3 ⋅ 3 ⋅ 3)

=  16 / 81

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