Using negative indices produces fractions. In this section we practice working with negative indices. From our work in the last section, we see that
a^{2} ÷ a^{3} | = | a^{2 – 3} | = | a^{–1} |
but we know that
a^{2} ÷ a^{3} | = | = | , a fraction. |
So clearly,
a^{–1} | = |
In same way,
a^{–2} | = | = |
a^{–3} | = | = |
and, in general,
a^{–n} | = |
for positive integer values of n. The three rules at the start of section 3.2 can now be used for any integers m and n, not just for positive values.
Calculate, leaving your answers as fractions:
(a) | 3^{–2} | |
(b) | 2^{–1} – 4^{–1} | |
(c) | 5^{–3} |
Simplify:
(a) | ||
(b) | 6^{4} × 6^{–3} | |
(c) | (10^{2})^{–3} |