IMAT Mathematics · Numbers, Powers and Logarithms

Powers, Fractional Exponents and Surds

The laws of powers let you rewrite any product of powers over one base; a fractional exponent is a root, and a surd in a denominator is removed with its conjugate.

All Numbers, Powers and Logarithms notes

Why this matters

Powers are asked almost every year: rewriting to a common base (2019, 2020, 2023), a root of a root (2024) and rationalising a denominator (2020). The 2023 question also needed a common factor taken out of a sum of powers.

Concept 1 of 3: Laws of powers and rewriting to a common base

A power is repeated multiplication, so multiplying two powers of the same base just counts the factors together. Most IMAT power questions mix bases like 4, 8 and 16; they become easy once everything is written as a power of the same prime (here 2).

Definition

For $a, b \ne 0$ and integers $m, n$ :

$a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ .
Same base: multiply means add exponents, divide means subtract, power of a power means multiply.
$(ab)^n = a^n b^n$ , but $(a + b)^n$ is not $a^n + b^n$ .
Common base: $4 = 2^2$ , $8 = 2^3$ , $16 = 2^4$ , $9 = 3^2$ , $27 = 3^3$ , $25 = 5^2$ .
If $a^p = a^q$ with $a > 0$ and $a \ne 1$ , then $p = q$ . This solves equations such as $9^x = 27$ .
A sum of powers does not combine; take out a common factor instead: $9^x + 3^x = 3^x(3^x + 1)$ .

Laws of powers

a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}

$a$ the base, the same in every factor
$m, n$ the exponents

Worked example

Write

\dfrac{27^n \times 9^{2n}}{3^n}

as a single power of 3.

Practice this conceptself-check · 4 quick reps

Adding equal powers does not add the exponents

2^{10} + 2^{10} = 2 \times 2^{10} = 2^{11}

, not

2^{20}

and not

4^{10}

. Exponents add only when powers of the same base are multiplied.

Concept 2 of 3: Fractional exponents and nth roots

Since

(a^{1/2})^2 = a^1 = a

, the power one half must be the square root. In the same way the power

\frac{1}{n}

is the nth root, and

\frac{m}{n}

is the nth root raised to the power m. The laws of powers still hold, so a root of a root is just a product of fractions.

Definition

For $a > 0$ :

$a^{1/n} = \sqrt[n]{a}$ , and $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ .
Take the root first: $32^{3/5} = (\sqrt[5]{32})^3 = 2^3$ is easier than $\sqrt[5]{32^3}$ .
A negative exponent means a reciprocal, not a negative number.
Root of a root: $\sqrt[m]{\sqrt[n]{a}} = a^{1/(mn)}$ .
An odd root of a negative number is real ( $\sqrt[3]{-8} = -2$ ); an even root of a negative number is not real.

Fractional exponent

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

$n$ the root (denominator of the exponent)
$m$ the power (numerator of the exponent)

Worked example

Evaluate (a)

32^{3/5}

, (b)

27^{-2/3}

, (c)

\sqrt[3]{\sqrt{64}}

Practice this conceptself-check · 4 quick reps

A negative exponent does not make the number negative

4^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}

, a positive number. The minus sign in an exponent means take the reciprocal.

Concept 3 of 3: Simplifying surds and rationalising the denominator

A surd is a root that cannot be simplified to a whole number, like

\sqrt{2}

. Pulling out the largest square factor makes surds comparable, so like surds can be collected. A surd in the denominator is cleared by using the difference of two squares:

(b + \sqrt{c})(b - \sqrt{c})

has no root left in it.

Definition

$\sqrt{ab} = \sqrt{a}\sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ for $a, b \ge 0$ .
Simplify by taking out the largest square factor: $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ .
Only like surds add: $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$ , but $\sqrt{2} + \sqrt{3}$ does not simplify.
Rationalise $\frac{a}{\sqrt{c}}$ by multiplying top and bottom by $\sqrt{c}$ .
Rationalise $\frac{a}{b + \sqrt{c}}$ by multiplying top and bottom by the conjugate $b - \sqrt{c}$ ; the denominator becomes $b^2 - c$ .

Rationalising with the conjugate

\frac{a}{b+\sqrt{c}} = \frac{a\,(b-\sqrt{c})}{b^2 - c}

$b - \sqrt{c}$ the conjugate of $b + \sqrt{c}$

Worked example

(a) Simplify

\sqrt{50} + \sqrt{18} - \sqrt{8}

. (b) Rationalise

\dfrac{6}{3 - \sqrt{5}}

Practice this conceptself-check · 4 quick reps

The root of a sum is not the sum of the roots

\sqrt{9 + 16} = \sqrt{25} = 5

, not

3 + 4 = 7

. Roots split over a product or a quotient, never over a sum or a difference.

The conjugate denominator is b² minus c

(b + \sqrt{c})(b - \sqrt{c}) = b^2 - c

. Writing

b^2 + c

is the usual slip, and an option built on it is always present.

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)

Laws of powers and rewriting to a common base
Laws of powers
$a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$
Fractional exponents and nth roots
Fractional exponent
$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$
Simplifying surds and rationalising the denominator
Rationalising with the conjugate
$\frac{a}{b+\sqrt{c}} = \frac{a\,(b-\sqrt{c})}{b^2 - c}$

Watch out for (4)

Adding equal powers does not add the exponents
→ Laws of powers and rewriting to a common base
A negative exponent does not make the number negative
→ Fractional exponents and nth roots
The root of a sum is not the sum of the roots
→ Simplifying surds and rationalising the denominator
The conjugate denominator is b² minus c
→ Simplifying surds and rationalising the denominator

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