E1.7 - Fractional indices (Extended) | Cambridge IGCSE International Mathematics

What could it mean to multiply 64 by itself “one third of a time”? Fractional indices extend the index laws you already know so that roots can be written as powers. This Extended lesson develops reciprocal, general fractional and negative fractional indices in E1.7, using positive bases throughout.

Roots as powers

The power $64^{1/3}$ must give 64 when cubed, because the power-of-a-power law multiplies the indices: $(64^{1/3})^3=64^1$ . It therefore means the cube root of 64, which is 4.

Reciprocal index

For a positive number $a$ and an integer $n\ge2$ , $a^{1/n}$ is the positive root of order $n$ of $a$ . Thus $a^{1/2}=\sqrt a$ , $a^{1/3}=\sqrt[3]a$ and $a^{1/4}=\sqrt[4]a$ .

The denominator tells you which root to take. An index of $1/3$ does not mean “divide the base by 3”.

Worked example 1: a square root as a power

Question. Work out $64^{1/2}$ .

Plan. Find the positive number whose square is 64.

\begin{aligned} 64^{1/2}&=\sqrt{64} &&\text{the denominator 2 means square root}\\ &=8 &&\text{because }8\times8=64 \end{aligned}

Answer. $8$ .

Check. $8^2=64$ . The root symbol asks for the positive root, so the answer is not $\pm8$ .

Keep the base fixed and change only the denominator. You are now looking for three equal factors rather than two.

Worked example 2: a cube root as a power

Question. Work out $64^{1/3}$ .

Plan. Find the positive number whose cube is 64.

\begin{aligned} 64^{1/3}&=\sqrt[3]{64} &&\text{the denominator 3 means cube root}\\ &=4 &&\text{because }4\times4\times4=64 \end{aligned}

Answer. $4$ .

Check. $(64^{1/3})^3=64^{(1/3)\times3}=64$ , and $4^3=64$ agrees.

A larger root index can give a smaller answer: for a base above 1, more equal factors need to be smaller.

Watch out

$64^{1/3}\ne64\div3$ . The fraction is an index; it changes the operation to a root. Check by raising your answer to the denominator's power.

Use that check with a fourth root.

General fractional powers

A fractional index with a numerator combines a root and a whole-number power. For $27^{2/3}$ , first take the cube root, then square the result. Taking the root first usually keeps the arithmetic small.

Fractional power

For $a>0$ , a positive integer $m$ and an integer $n\ge2$ ,

a^{m/n}=\left(\sqrt[n]a\right)^m=\sqrt[n]{a^m}.

The denominator gives the root; the numerator gives the power. This rule is not on the supplied formula list: learn it.

The two orders agree by the power-of-a-power law. For example, $(27^{1/3})^2=27^{(1/3)\times2}$ .

Worked example 3: root, then square

Question. Work out $27^{2/3}$ without a calculator.

Plan. Take the cube root before squaring.

\begin{aligned} 27^{2/3}&=\left(\sqrt[3]{27}\right)^2 &&\text{denominator 3, numerator 2}\\ &=3^2 &&\text{since }3^3=27\\ &=9 \end{aligned}

Answer. $9$ .

Check. Cube the answer: $9^3=729=27^2$ , as the index law requires.

Now change the root and the power. Keep the same order of decisions: read the denominator before the numerator.

Worked example 4: root, then cube

Question. Work out $16^{3/4}$ without a calculator.

Plan. Take the fourth root, then cube it.

\begin{aligned} 16^{3/4}&=\left(\sqrt[4]{16}\right)^3\\ &=2^3 &&\text{since }2^4=16\\ &=8 \end{aligned}

Answer. $8$ .

Check. $8^4=4096$ and $16^3=4096$ . Raising to the denominator gives the numerator's power of the base.

You can simplify an index fraction first. For example, $a^{2/4}=a^{1/2}$ ; both describe the same positive-base power.

Watch out

$16^{3/4}$ is not $\sqrt[3]{16^4}$ . That swaps the numerator and denominator. Read it as “fourth root, then cube”.

Use a fifth root in the next calculation.

Combining index laws

A negative fractional index has two instructions: the fraction gives a root and power, and the minus sign gives a reciprocal. It does not make a positive base's value negative.

Negative fractional power

For $a>0$ ,

a^{-m/n}=\frac{1}{a^{m/n}}=\frac{1}{(\sqrt[n]a)^m}.

The usual index laws still apply: multiply powers of the same base by adding indices, divide by subtracting indices, and raise a power to a power by multiplying indices. These rules are not on the supplied formula list: learn them.

Choose whether to simplify the indices first or evaluate the roots first. Both routes must preserve the same base.

Worked example 5: a negative root power

Question. Work out $25^{-1/2}$ without a calculator.

Plan. Evaluate the positive fractional power and take its reciprocal.

\begin{aligned} 25^{-1/2}&=\frac{1}{25^{1/2}} &&\text{negative index means reciprocal}\\ &=\frac{1}{\sqrt{25}}\\ &=\frac15 \end{aligned}

Answer. $\frac15$ .

Check. $\frac15\times25^{1/2}=\frac15\times5=1$ : the powers with opposite indices are reciprocals.

The next expression produces a negative fractional index after combining two powers. Add the indices before finding the root.

Worked example 6: combine, then evaluate

Question. Work out $27^{2/3}\times27^{-1}$ without a calculator.

Plan. Add the indices because the bases are the same.

\begin{aligned} 27^{2/3}\times27^{-1}&=27^{2/3-1}\\ &=27^{-1/3} &&\text{write }1=3/3\\ &=\frac{1}{\sqrt[3]{27}}\\ &=\frac13 \end{aligned}

Answer. $\frac13$ .

Check. Evaluate separately: $27^{2/3}=9$ and $27^{-1}=1/27$ , so their product is $9/27=1/3$ .

The opening “one third of a time” now has a precise meaning: the value that cubes back to the original positive number. The index laws led us to roots, rather than to a new multiplication rule.

Watch out

When dividing, subtract the entire second index. For example, $a^{1/2}\div a^{-1/2}=a^{1/2-(-1/2)}=a^1$ , not $a^0$ , for $a>0$ .

Keep the subtraction visible in the next calculation.

Quick Recap

A reciprocal index $1/n$ means a root of order $n$ .
In $a^{m/n}$ , take the root from the denominator and the power from the numerator.
A negative index means a reciprocal; it does not mean a negative value.
Index laws work with fractional indices for the positive bases used here.

In particular, $64^{1/3}=4$ because $4^3=64$ .

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E1.7 - Fractional indices (Extended)

Roots as powers

General fractional powers

Combining index laws

Quick Recap

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