AFCAT Numerical Ability · Indices and Surds

Fractional Powers, Roots as Powers, and Equations in Powers

Read a fractional or decimal index as a root and a power, write every root as an index, and solve an equation by putting both sides on one base.

All Indices and Surds notes

Why this matters

This is the HARD end of the chapter's index questions: about two in five here are HARD, against a handful on the laws page. They are decimal powers such as a power of 0.25 or 1.5, products of two numbers raised to different decimals, and nested roots. Each one falls to the same two moves: write the base as a power of a prime, and write the index as a fraction.

Concept 1 of 4: A root is a fractional power

a^{1/2}

multiplied by itself is

a^{1/2 + 1/2} = a

, so

a^{1/2}

must be the square root. In the same way

a^{1/3}

is the cube root and

a^{1/n}

is the

n

th root. Once a root is a power, the laws of exponents handle it.

Definition

$\sqrt[n]{a} = a^{1/n}$ , and $\sqrt{a} = a^{1/2}$ .
To find $a^{1/n}$ , write $a$ as a perfect $n$ th power: $32 = 2^5$ , so $32^{1/5} = 2$ .
A root raised to a power: $\left(\sqrt[n]{a}\right)^k = a^{k/n}$ .
A root of a root multiplies the indices: $\sqrt[3]{\sqrt{a}} = a^{1/6}$ .

Root as an index

\sqrt[n]{a} = a^{1/n}, \qquad \sqrt[m]{\sqrt[n]{a}} = a^{1/(mn)}

Worked example

Find

125^{1/3}

and

\sqrt[3]{\sqrt{4096}}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Indices and Surds · Fractional Powers and Roots

Simplify:

\sqrt[4]{81}

A half power is a square root, not half

36^{1/2} = 6

, not

18

. The index

\tfrac 12

asks for the number that squares to 36.

Nested roots multiply their indices

\sqrt[3]{\sqrt{x}} = x^{1/6}

, not

x^{1/5}

. A root of a root is a power of a power, so the indices multiply.

Drill 5 more on a root is a fractional power

Concept 2 of 4: Powers like m/n and decimal powers

a^{m/n}

is two steps: the

n

th root, then the

m

th power. Take the root first and the numbers stay small. A decimal index is only a fraction in disguise, so turn it into one before you start.

Definition

$a^{m/n} = \left(\sqrt[n]{a}\right)^m$ .
Decimals to fractions: $0.25 = \tfrac 14$ , $0.5 = \tfrac 12$ , $0.75 = \tfrac 34$ , $1.5 = \tfrac 32$ , $2.5 = \tfrac 52$ , $0.6 = \tfrac 35$ .
A whole-plus-half index leaves a square root: $a^{5/2} = a^2 \sqrt a$ .
Two different bases with awkward decimals: rewrite both on one prime base, then add the indices. $9^{0.3} \times 27^{0.8}$ is $3^{0.6} \times 3^{2.4}$ .
A sum such as $a^{m/n} + b^{p/q}$ : find each power on its own, then add.

Fractional index

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

Worked example

Find

32^{0.6}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Indices and Surds · Fractional Powers and Roots

Evaluate:

(256)^{0.16} \times (16)^{0.18}

A decimal index is a fraction, not a divisor

x^{0.25}

x^{1/4}

, the fourth root of

x

. It is not

x

divided by 4, and

625^{0.25} = 5

, not

156.25

Same index: multiply bases. Different indices: one base

8^{0.2} \times 4^{0.2} = 32^{0.2} = 2

, because the indices match. With

8^{0.2} \times 4^{0.3}

they do not, so rewrite as

2^{0.6} \times 2^{0.6} = 2^{1.2}

Root first keeps the numbers small

1000^{2/3}

: the cube root is 10 and its square is 100. Squaring first means a cube root of a million, which is the same answer the long way.

Drill 10 more on powers like m/n and decimal powers

Concept 3 of 4: Fractional indices on letters

The laws do not care whether the index is a whole number.

\dfrac{x^{5/6}}{x^{1/3}}

is still top index minus bottom index; the only extra work is subtracting fractions over a common denominator.

Definition

Write every root as an index first: $\sqrt{a^3} = a^{3/2}$ and $(\sqrt a)^3 = a^{3/2}$ as well.
Then use the laws: add for a product, subtract for a quotient, multiply for a power of a power.
Bring fractions to one denominator before adding or subtracting them: $\tfrac 56 - \tfrac 13 = \tfrac 56 - \tfrac 26 = \tfrac 12$ .
A letter whose indices cancel to 0 becomes 1.

Quotient with fractional indices

\dfrac{x^{a/b}}{x^{c/d}} = x^{\frac ab - \frac cd}

Worked example

Simplify

\dfrac{p^{5/6}\, q^{4/5}}{p^{1/3}\, q^{2/5}}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Indices and Surds · Fractional Powers and Roots

Simplify:

\frac{x^{1/2} \times y^{1/3}}{x^{1/4} \times y^{1/6}}

The root of a cube is a three-halves power

\sqrt{a^3} = a^{3/2}

, not

a^{3}/2

and not

a^{6}

. The root divides the index by 2.

Subtract fractions over one denominator

\tfrac 45 - \tfrac 13

\tfrac{12 - 5}{15} = \tfrac{7}{15}

. Subtracting tops and bottoms separately gives

\tfrac 32

, which is wrong.

Drill 3 more on fractional indices on letters

Concept 4 of 4: Equations in powers and in square roots

2^x = 2^5

, then

x = 5

: equal powers of the same base have equal indices. So rewrite both sides on one base and compare indices. An equation with a square root is the other way round: get the root alone, then square it away.

Definition

Unknown in the index: write both sides as powers of one prime, then equate indices. A square root on the right is a half power: $9\sqrt 3 = 3^{2.5}$ .
Unknown under a root: move everything else to the other side first, then square both sides.
Put the answer back into the original equation. Squaring can create a value that does not fit.

Equal powers of one base

a^{x} = a^{k} \;\Rightarrow\; x = k \quad (a > 0,\ a \ne 1)

Worked example

Solve

4^{x+1} = 32

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Indices and Surds · Fractional Powers and Roots

Solve:

2^x = 8\sqrt{2}

Isolate the root before squaring

\sqrt{x + 1} + 2 = 5

, first get

\sqrt{x + 1} = 3

, so

x = 8

. Squaring the left side as it stands does not remove the root.

Match the bases before matching indices

4^x = 8

does not give

x = 2

. Write it as

2^{2x} = 2^3

, so

x = 1.5

Drill 3 more on equations in powers and in square roots

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 (4)

A root is a fractional power
Root as an index
$\sqrt[n]{a} = a^{1/n}, \qquad \sqrt[m]{\sqrt[n]{a}} = a^{1/(mn)}$
Powers like m/n and decimal powers
Fractional index
$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$
Fractional indices on letters
Quotient with fractional indices
$\dfrac{x^{a/b}}{x^{c/d}} = x^{\frac ab - \frac cd}$
Equations in powers and in square roots
Equal powers of one base
$a^{x} = a^{k} \;\Rightarrow\; x = k \quad (a > 0,\ a \ne 1)$

Watch out for (9)

A half power is a square root, not half
→ A root is a fractional power
Nested roots multiply their indices
→ A root is a fractional power
A decimal index is a fraction, not a divisor
→ Powers like m/n and decimal powers
Same index: multiply bases. Different indices: one base
→ Powers like m/n and decimal powers
Root first keeps the numbers small
→ Powers like m/n and decimal powers
The root of a cube is a three-halves power
→ Fractional indices on letters
Subtract fractions over one denominator
→ Fractional indices on letters
Isolate the root before squaring
→ Equations in powers and in square roots
Match the bases before matching indices
→ Equations in powers and in square roots

Drill the questions on this topic

25 questions from the bank, with cart and Word export.

Drill all 25 questions