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Simplifying expressions using the laws of indicesNegative indices

Indices show where a number has been multiplied by itself, eg squared or cubed, or to show roots of numbers, eg square root. Some terms with indices can be simplified using the laws of indices.

Part of MathsNumerical skills

Negative indices

Example

\(\frac{{{y^4}}}{{{y^5}}}\)

\(= {y^{4 - 5}}\)

\(= {y^{ - 1}}\)

If we write this out in full, we get:

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

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

So \(= {y^{ - 1}}\) means \(\frac{1}{y}\)

In general we get \({a^{ - n}} = \frac{1}{{{a^n}}}\)

For example \(a^{-2}=\frac{1}{a^{2}}\), \(a^{-5}=\frac{1}{a^{5}}\)

Question

Simplify \(\frac{{{y^7}}}{{{y^9}}}\)

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