AFCAT Numerical Ability · Squares, Cubes and Roots

Roots as Fractional Powers, and the Laws of Indices

Write a root as a fractional power and back, combine powers of one base with the laws of indices, compare roots of different kinds, and solve equations where powers are equal.

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Why this matters

Most of this page turns a root into a power, or a power into a root, and then uses the laws of indices. The harder questions set several powers equal to one number. Marks go on reading the top of a fractional power as the root, and on multiplying indices that should be added.

Concept 1 of 3: A root as a fractional power

A fractional power is a root and a power in one: the bottom of the fraction says which root, the top says which power. A square root is the power one half, a cube root the power one third.

Definition

$a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ .
Take the root first when the number is a perfect power, so the numbers stay small: $27^{2/3} = 3^2 = 9$ .
For whole numbers, $\sqrt[n]{m}$ is a whole number when $m$ is a perfect $n$ th power, and irrational otherwise: $\sqrt[3]{16}$ is irrational.

Fractional power

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

Worked example

Find

81^{3/4}

and

25^{3/2}

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 1 · Squares, Cubes and Roots · Fractional Powers and Indices

Evaluate:

4^{3/2} - 8^{2/3}

The bottom is the root

x^{2/3}

is the cube root of

x^2

, not the square root of

x^3

, which is

x^{3/2}

A half power is not half

36^{1/2} = 6

, not 18. The power one half takes a square root.

Drill 7 more on a root as a fractional power

Concept 2 of 3: The laws of indices

Powers of the same base combine by their indices: multiplying adds them, dividing subtracts them, and a power of a power multiplies them. With roots written as fractional powers, the same laws handle roots.

Definition

$a^m \times a^n = a^{m + n}$ , $a^m \div a^n = a^{m - n}$ and $(a^m)^n = a^{mn}$ .
$a^0 = 1$ and $a^{-n} = \dfrac{1}{a^n}$ .
A root inside a root multiplies the indices: $\sqrt{\sqrt[3]{a}} = a^{1/6}$ .
To compare roots of different kinds, raise them all to the LCM of the roots. For $\sqrt 2$ and $\sqrt[3]{3}$ , the 6th powers are 8 and 9, so $\sqrt[3]{3}$ is the larger.
Halving again and again is a power of 2: halving 6 times multiplies by $2^{-6}$ .

Laws of indices

a^m a^n = a^{m + n}, \quad \dfrac{a^m}{a^n} = a^{m - n}, \quad (a^m)^n = a^{mn}

Worked example

Simplify

\sqrt{5\sqrt{5\sqrt 5}}

to a single power of 5.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Squares, Cubes and Roots · Fractional Powers and Indices

\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}

is equal to

Add, do not multiply

x^{1/2} \times x^{1/4} = x^{3/4}

, not

x^{1/8}

. Multiplying powers of one base adds the indices.

A negative index is not a negative number

5^{-2} = \tfrac{1}{25}

, a positive number, not

-25

Drill 4 more on the laws of indices

Concept 3 of 3: Equations with equal powers

When two powers of the same base are equal, their indices are equal. When several powers all equal one number

k

, write each base as a power of

k

, and a product of bases becomes a sum of indices.

Definition

$a^x = a^y$ means $x = y$ (for $a > 0$ , $a \neq 1$ ). Write both sides in one base first: $2^{0.5x} = 32 = 2^5$ gives $x = 10$ .
If $a^x = b^y = k$ , then $a = k^{1/x}$ and $b = k^{1/y}$ , so $ab = k^{1/x + 1/y}$ and $\dfrac ab = k^{1/x - 1/y}$ .
Build a new number out of the known ones: if $5^p = 2$ and $5^q = 3$ , then $6 = 2 \times 3$ gives $5^{p + q} = 6$ .

Equal powers

a^x = b^y = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y}

Worked example

2^x = 3^y = 6^z

, find

z

in terms of

x

and

y

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Squares, Cubes and Roots · Fractional Powers and Indices

a^x = b^y = c^z

and

abc = 1

, then what is

xy + yz + zx

equal to

Same base first

4^x = 2^6

does not give

x = 6

. Write 4 as

2^2

2^{2x} = 2^6

, so

x = 3

A product becomes a sum

a = k^{1/x}

and

b = k^{1/y}

, then

ab = k^{1/x + 1/y}

, not

k^{1/(xy)}

Drill 5 more on equations with equal 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 (3)

A root as a fractional power
Fractional power
$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$
The laws of indices
Laws of indices
$a^m a^n = a^{m + n}, \quad \dfrac{a^m}{a^n} = a^{m - n}, \quad (a^m)^n = a^{mn}$
Equations with equal powers
Equal powers
$a^x = b^y = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y}$

Watch out for (6)

The bottom is the root
→ A root as a fractional power
A half power is not half
→ A root as a fractional power
Add, do not multiply
→ The laws of indices
A negative index is not a negative number
→ The laws of indices
Same base first
→ Equations with equal powers
A product becomes a sum
→ Equations with equal powers

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