Formula sheet

Fractional index

Q: Which number is the root and which is the power?

In am/n the denominator n is the root and the numerator m is the power: am/n is the n-th root of am. So for 82/3 you take the cube root (n = 3) and square it (m = 2): (³√8)² = 2² = 4. Bottom is the root, top is the power.

Q: Does a 1/2 really mean the square root?

Yes. Putting n = 2 and m = 1 gives a1/2 = ²√a¹ = √a. A fractional index of one over a number is simply that root: a1/3 is the cube root, a1/4 is the fourth root, and so on for larger denominators. The numerator of 1 leaves the power as one.

Q: Can I take the power first, then the root?

Yes, the order does not change the answer. 163/4 can be worked as (⁴√16)³ = 2³ = 8, or as ⁴√(16³) = ⁴√4096 = 8, both give 8. Taking the root before the power usually keeps the numbers smaller and much easier to handle by hand.

The meaning of a fractional index is given on the formula sheet. What you must know is how to read it: the denominator n is the root and the numerator m is the power, and that taking the root first usually keeps the numbers small and manageable.

Written by the spmmath.com.my editorial team.

Fractional index

a^m/n = (n-th root of a^m)

Given in the exam

What the symbols mean

a the base, the positive number being rooted and powered
m the numerator of the fractional index, the power
n the denominator of the fractional index, the root (n-th root)

Given in the exam, or memorise?

Why it works

Fractional indices are defined so the ordinary index laws still hold.

If a^1/n is raised to the power n, the indices multiply to give a^n/n = a.
So a^1/n must be the number whose n-th power is a, the n-th root of a.
Then a^m/n = (a^1/n)^m, i.e. the n-th root of a raised to the power m, which is the n-th root of a^m.

Worked example 1

Evaluate 8^2/3.

Read the fractional index: denominator 3 is the root and numerator 2 is the power, so 8^2/3 is the cube root of 8².
Take the cube root first to keep the numbers small: ³√8 = 2.
Raise the result to the power 2: 2² = 4.

Worked example 2

Evaluate 16^3/4.

Denominator 4 is the root and numerator 3 is the power: 16^3/4 is the fourth root of 16³.
Take the fourth root first: ⁴√16 = 2, since 2⁴ = 16.
Raise the result to the power 3: 2³ = 8.

Where students go wrong

Swapping m and n, the root is the denominator n and the power is the numerator m, not the other way round.
Multiplying the base by the fraction, 8^2/3 is not 8 × 2/3; it is a root and a power, giving 4.
Forgetting that a^1/2 = √a, or taking the root of only part of the expression.

Use it with

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

Which number is the root and which is the power?

In a^m/n the denominator n is the root and the numerator m is the power: a^m/n is the n-th root of a^m. So for 8^2/3 you take the cube root (n = 3) and square it (m = 2): (³√8)² = 2² = 4.

Bottom is the root, top is the power.

Does a^1/2 really mean the square root?

Yes. Putting n = 2 and m = 1 gives a^1/2 = ²√a¹ = √a.

A fractional index of one over a number is simply that root: a^1/3 is the cube root, a^1/4 is the fourth root, and so on for larger denominators. The numerator of 1 leaves the power as one.

Can I take the power first, then the root?

Yes, the order does not change the answer. 16^3/4 can be worked as (⁴√16)³ = 2³ = 8, or as ⁴√(16³) = ⁴√4096 = 8, both give 8.

Taking the root before the power usually keeps the numbers smaller and much easier to handle by hand.