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Cambridge IGCSE Mathematics — 0580 Extended

Topic 1.6: Number — Indices

What an index means

An index (power) tells you how many times a base is multiplied by itself.

\[ a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}} \]

For example, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). The base is \(2\); the index is \(4\).

Index laws (same base)

  1. Multiply: \(a^m \times a^n = a^{m+n}\)
  2. Divide: \(a^m \div a^n = a^{m-n}\)
  3. Power of a power: \((a^m)^n = a^{mn}\)

Zero and negative indices

Zero index: for any \(a \neq 0\):

\[ a^0 = 1 \]

Negative index: a negative index means “take the reciprocal”:

\[ a^{-n} = \dfrac{1}{a^n} \]

Worked example 1

Evaluate \(2^{-3}\).

Worked solution: two to the power negative three equals one eighth
Negative index: \(2^{-3} = \dfrac{1}{8}\)

Worked example 2

Simplify \(2^{-3} \times 2^4\).

Worked solution: two to the negative three times two to the four equals two
Index law: \(2^{-3} \times 2^4 = 2\)

Paper 2 (non-calculator)

When indices have different bases, rewrite using prime factors first. Example: \(4^{-2} \times 2^3 = (2^2)^{-2} \times 2^3 = 2^{-4} \times 2^3 = 2^{-1} = \dfrac{1}{2}\).

Fractional indices

The denominator of the fraction is the root:

\[ a^{\frac{1}{n}} = \sqrt[n]{a} \]

Find the root first, then raise to the power:

\[ a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m = \left(\sqrt[n]{a}\right)^m \]

Worked example 3

Evaluate \(81^{\frac{1}{2}}\).

Worked solution: eighty-one to the power one half equals nine
Fractional index: \(81^{\frac{1}{2}} = 9\)

Worked example 4

Evaluate \(8^{\frac{2}{3}}\).

Worked solution: eight to the power two thirds equals four
Fractional index: \(8^{\frac{2}{3}} = 4\)

Which method?

Same base, multiplying or dividing Use index laws — add or subtract the powers.
Fractional index with a perfect power Take the root (denominator), then apply the power (numerator).
Different bases Write each base as a product of prime factors, then apply the laws.

Try this

Simplify \(3^{-2} \times 3^5\) and evaluate \(27^{\frac{1}{3}}\).

Show answer
Answer

  1. Same base — add the indices

    \[ 3^{-2} \times 3^{5} = 3^{-2+5} = 3^{3} \]
  2. \[ 3^{3} = 27 \]

  3. Fractional index — cube root

    \[ 27^{\frac{1}{3}} = \sqrt[3]{27} \]

  4. Answer

    \[ 3^{-2} \times 3^{5} = 27,\quad 27^{\frac{1}{3}} = 3 \]

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