Law 3: Negative Indices (Powers)

This law applies when the power is negative.

\(\scriptsize a^{-n} = \normalsize \frac{1}{a^n}\)

In other words, an index number with a negative power is equal to the inverse of its index number with positive power. 

\(\scriptsize 3^{-3}= \normalsize \frac{1}{3^{3}}\)

\(\scriptsize 2^{-4}= \normalsize \frac{1}{2^{4}}\)

Worked Example 5.3.1:

Simplify the following:

a. \( \scriptsize 2^4 \: \div \: 2^8 \)

b. y²  ÷  y⁶

c. x⁵ ÷ x⁷

Solution

a. \( \scriptsize 2^4 \: \div \: 2^8 \\= \frac{\not{2} \: \times \: \not{2} \: \times \: \not{2}\: \times \: \not{2}}{\not{2} \: \times \: \not{2} \: \times \: \not{2}\: \times \: \not{2}\: \times \: 2\: \times \: 2\: \times \: 2 \: \times \: 2} \\= \frac{1}{2^4}\)

Alternatively, using the law:

\( \scriptsize 2^4 \: \div \: 2^8 \\ \scriptsize = 2^{4 \: – \: 8} \\ \scriptsize = 2^{-4} \)

= \( \frac{1}{2^4}\)

b. \( \scriptsize y^2 \: \div \: y^6 \\ = \frac{\not{y} \: \times \: \not{y}}{\not{y}\: \times \: \not{y} \: \times \: y \: \times \: y \: \times \: y\: \times \: y} \\= \frac{1}{y^4}\)

Alternatively, using the law:

\( \scriptsize y^2 \: \div \: y^6 \\ \scriptsize = y^{2 \: – \: 6} \\ \scriptsize = y^{-4} \)

= \( \frac{1}{y^4}\)

c. \( \scriptsize x^5 \: \div \: x^7 \\= \frac{ \not{x} \: \times \: \not{x}\: \times \: \not{x} \: \times \: \not{x} \: \times \: \not{x}}{\not{x} \: \times \: \not{x} \: \times \: \not{x}\: \times \: \not{x}\: \times \: \not{x}\: \times \: x\: \times \: x } \\= \frac{1}{x^2}\)

Alternatively, using the law:

\( \scriptsize x^5 \: \div \: x^7 \\ \scriptsize = x^{5 \: – \: 7} \\ \scriptsize = x^{-2} \)

= \( \frac{1}{x^2}\)

Worked Example 5.3.2:

Simplify the following;

a. \( \scriptsize 5^{-2} \)

b. \( \scriptsize 3^{-5} \)

c. \( \left(\frac{2}{3} \right)^{-1} \)

d. \( \left(\frac{4}{5} \right)^{-2} \)

e. \( \scriptsize 2^5 \: \times \: 2^{-3} \: \times \: 2^{-8} \)

f. \( \scriptsize \left( \scriptsize 5y \right)^{-2} \)

g. \( \scriptsize 5y^{-2} \)

h. \( \scriptsize 5^{-2}y \)

Solution

a. \( \scriptsize 5^{-2} \) (negative index)

= \( \frac{1}{5^2} \) (positive index)

b. \( \scriptsize 3^{-5} \) (negative index)

= \( \frac{1}{3^5} \) (positive index)

c. \( \left(\frac{2}{3} \right)^{-1} \) (negative index)

= \(\frac{1}{\frac{2}{3}} \)

= \( \scriptsize 1 \: \div \: \frac{2}{3} \)

= \( \scriptsize 1 \: \times\: \frac{3}{2} \)

= \( \frac{3}{2} \) (positive index)

d. \( \left(\frac{4}{5} \right)^{-2} \) (negative index)

= \(\frac{1}{\left(\frac{4}{5}\right)^2} \)

= \( \scriptsize 1 \: \div \: \left(\frac{4}{5}\right)^2 \)

= \( \scriptsize 1 \: \times\: \left(\frac{5}{4}\right)^2 \)

= \( \left(\frac{5}{4}\right)^2 \) (positive index)

= \( \frac{25}{16} \)

e. \( \scriptsize 2^5 \: \times \: 2^{-3} \: \times \: 2^{-8} \) (same base)

= \( \scriptsize 2^{5 +(-3)+(-8)}\)

= \( \scriptsize 2^{5 -3-8}\)

= \( \scriptsize 2^{5 -11}\)

= \( \scriptsize 2^{-6}\)

=\(\frac{1}{2^6}\)

f. \( \scriptsize \left( \scriptsize 5y \right)^{-2} \)

Note: 5y is enclosed in the bracket. The index affects both 5 and y.

=\( \frac{1}{(5y)^2} \)

= \( \frac{1}{5y \: \times \: 5y} \)

= \( \frac{1}{25y^2} \)

g. \( \scriptsize \scriptsize 5y^{-2} \)

Note: The negative index affects only y in this case

= \(\scriptsize 5 \: \times \: y^{-2} \)

= \(\scriptsize 5 \: \times \: \normalsize \frac{1}{y^2} \)

= \( \frac{5}{y^2} \)

h. \( \scriptsize \scriptsize 5^{-2}y \)

Note: The negative index affects only 5 in this case

= \(\scriptsize 5^{-2} \: \times \: y\)

= \( \frac{1}{5^2} \scriptsize \: \times \: y \)

= \( \frac{1}{25} \scriptsize \: \times \: y \)

= \( \frac{y}{25} \)