SPI Mathematics · Surds, Indices and Simplification

Fractional Indices and Exponential Equations

A fractional index is a root followed by a power, and an equation in powers is solved by writing both sides as powers of one base and matching the indices.

All Surds, Indices and Simplification notes

Why this matters

Eleven questions in the bank: three EASY, seven MODERATE and the chapter's one HARD question. Seven ask for the value of a fractional or negative power, and four ask you to solve an equation by writing both sides as powers of the same base.

Concept 1 of 2: Fractional indices as roots

Squaring and taking a square root undo each other, so

a^{\frac{1}{2}}

must be

\sqrt{a}

. In

a^{\frac{p}{q}}

the bottom number is the root and the top number is the power. Write the base as a power first and the arithmetic stays small.

Definition

$a^{\frac{1}{n}} = \sqrt[n]{a}$ , the $n$ th root of $a$ .
$a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^p$ : take the $q$ th root, then raise to the power $p$ .
First write the base as a power: $81 = 3^4$ , $\frac{243}{32} = \left(\frac{3}{2}\right)^5$ , $0.008 = (0.2)^3$ . Then multiply the indices.
A negative fractional index: turn the base over as well, $\left(\frac{4}{9}\right)^{-\frac{1}{2}} = \left(\frac{9}{4}\right)^{\frac{1}{2}} = \frac{3}{2}$ .
An odd root of a negative number is negative: $\sqrt[3]{-8} = -2$ .

Fractional index

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

Worked example

Find

81^{\frac{3}{4}}

Practice this conceptself-check · 5 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Surds, Indices and Simplification · Fractional Indices and Exponential Equations

\left(\frac{243}{32}\right)^{-\frac{4}{5}} =

Swapping the root and the power

8^{\frac{2}{3}}

the 3 is the root and the 2 the power:

(\sqrt[3]{8})^2 = 4

. Reading it as

8^{\frac{3}{2}}

gives about 22.6.

Forgetting to turn a fraction over

\left(\frac{9}{16}\right)^{-\frac{1}{2}}

\frac{4}{3}

, not

\frac{3}{4}

. A negative index flips the fraction before the root is taken.

Miscounting the decimal places

0.0016 = (0.2)^4

because

2^4 = 16

and four places of 0.1 give four decimal places. Count places before writing a decimal as a power.

Drill 6 more on fractional indices as roots

Concept 2 of 2: Solving equations in powers

If two powers of the same base are equal, their indices must be equal too. So the whole job is to rewrite each side as a power of one base: then the unknown comes out of a plain equation.

Definition

If $a^x = a^y$ with $a > 0$ and $a \ne 1$ , then $x = y$ .
Rewrite both sides with one base: $9 = 3^2$ , $27 = 3^3$ , $\frac{1}{4} = 2^{-2}$ .
When powers are added or subtracted, take out the smaller power as a common factor: $2^{x+3} - 2^{x+1} = 2^{x+1}(2^2 - 1)$ .
When several letters are linked, write each one as a power of the same letter, then compare indices.

Equal powers of one base

a^x = a^y \Rightarrow x = y \quad (a > 0,\ a \ne 1)

Worked example

Solve

2^{x+3} - 2^{x+1} = 48

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Surds, Indices and Simplification · Fractional Indices and Exponential Equations

The value of x, if

5^{2x - 1} = 25^{x - 1} + 100

is ____.

Matching indices on different bases

From

4^x = 8

you cannot say

x = 2

. Make the bases equal first,

2^{2x} = 2^3

, and only then match the indices.

Treating a difference of powers as one power

3^{x+2} - 3^x

is not

3^2

. Take out

3^x

: it equals

3^x(9 - 1) = 8 \times 3^x

Missing the answer x = 1

Matching indices needs a base that is not 1. In

x^{2x} = x^6

the indices give

x = 3

, but

x = 1

also works, because 1 to any power is 1. Check which values the question allows.

Drill 3 more on solving equations in powers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Fractional indices as roots
Fractional index
$a^{\frac{1}{n}} = \sqrt[n]{a} \qquad a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^{p}$
Solving equations in powers
Equal powers of one base
$a^x = a^y \Rightarrow x = y \quad (a > 0,\ a \ne 1)$

Watch out for (6)

Swapping the root and the power
→ Fractional indices as roots
Forgetting to turn a fraction over
→ Fractional indices as roots
Miscounting the decimal places
→ Fractional indices as roots
Matching indices on different bases
→ Solving equations in powers
Treating a difference of powers as one power
→ Solving equations in powers
Missing the answer x = 1
→ Solving equations in powers

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