Indices and Surds

If a bacteria count doubles every hour, after 10 hours it is $2^{10}$ times as large — that is an index: a short way to write repeated multiplication. On the other side are numbers like $2$ whose exact value never ends as a decimal; these are surds, and in the exam you must simplify, add, compare and rationalise them without a calculator.

The chapter has two halves: indices (laws, fractional and negative powers, equations) and surds (spotting, operations, comparing, rationalising, roots inside roots).

1

Indices

Laws · fractional and negative powers · equations

The seven laws, how to tame hard-looking expressions, and exponent equations.

Laws of indices

Same base — work on the powers

What $a^{n}$ means and the seven laws.

In $a^{n}$ , $a$ is the base and $n$ the index: multiply $a$ by itself $n$ times: $2^{4} = 2 \times 2 \times 2 \times 2 = 16$ .

Law	Example
$a^{m} \times a^{n} = a^{m + n}$	$2^{3} \times 2^{4} = 2^{7}$
$a^{m} \div a^{n} = a^{m - n}$	$5^{7} \div 5^{4} = 5^{3}$
$(a^{m})^{n} = a^{mn}$	$(3^{2})^{4} = 3^{8}$
$(ab)^{n} = a^{n} b^{n}$	$(2 \times 5)^{3} = 8 \times 125$
$(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$	$(\frac{2}{3})^{2} = \frac{4}{9}$
$a^{0} = 1 (a \neq = 0)$	$7^{0} = 1$
$a^{- n} = \frac{1}{a ^{n}}$	$2^{- 3} = \frac{1}{8}$

CarefulAdd vs power:

a^{m} + a^{n} \neq = a^{m + n}

and

(a + b)^{n} \neq = a^{n} + b^{n}

. The laws work only on multiplication / division.

Example 1 · Basic

Evaluate

\frac{2 ^{5} \times 2 ^{3}}{2 ^{4}}

.

Solution, step by step

1Multiplying adds the powers: $2^{5 + 3} = 2^{8}$ .
2Dividing subtracts: $2^{8 - 4} = 2^{4} = 16$ .

Answer 16

Example 2 · Standard

Evaluate

\frac{4 ^{3} \times 8 ^{2}}{1 6 ^{2}}

.

Solution, step by step

1Write all as powers of 2: $4 = 2^{2}$ , $8 = 2^{3}$ , $16 = 2^{4}$ .
2 $\frac{2 ^{6} \times 2 ^{6}}{2 ^{8}} = 2^{12 - 8} = 2^{4} = 16$

Answer 16

BasicTry it yourself first

What is the value of

\frac{3 ^{5} \times 3 ^{2}}{3 ^{4}}

?

BasicTry it yourself first

What is the value of

\frac{5 ^{6} \times 5 ^{3}}{5 ^{7}}

?

StandardTry it yourself first

What is the value of

\frac{2 ^{7} \times 2 ^{9}}{2 ^{12}}

?

StandardTry it yourself first

What is the value of

\frac{7 ^{4} \times 7 ^{3}}{7 ^{8}}

?

StandardTry it yourself first

What is the value of

\frac{4 ^{4} \times 8 ^{2}}{1 6 ^{3}}

?

StandardTry it yourself first

What is the value of

\frac{4 ^{5} \times 8 ^{3}}{1 6 ^{6}}

?

AdvancedTry it yourself first

What is the value of

\frac{4 ^{7} \times 8 ^{5}}{1 6 ^{9}}

?

Example 3 · Standard

Evaluate

\frac{3 ^{n + 2} - 3 ^{n}}{3 ^{n - 1}}

.

Solution, step by step

1Take $3^{n}$ common in the numerator: $3^{n} (3^{2} - 1) = 3^{n} \times 8$ .
2The denominator $3^{n - 1} = 3^{n} \div 3$ , so the value is $8 \times 3 = 24$ .

Answer 24

The answer does not depend on

n

.

StandardTry it yourself first

What is the value of

\frac{2 ^{n + 3} - 2 ^{n}}{2 ^{n - 2}}

?

StandardTry it yourself first

What is the value of

\frac{5 ^{n + 2} - 5 ^{n}}{5 ^{n - 1}}

?

AdvancedTry it yourself first

What is the value of

\frac{3 ^{n + 4} - 3 ^{n}}{3 ^{n - 3}}

?

Fractional and negative powers

The link with roots

$a^{1/ n}$ means the $n$ th root; how to read $a^{m / n}$ and $a^{- n}$ .

a^{1/ n} = n a, a^{m / n} = (n a)^{m}, a^{- m / n} = \frac{1}{a ^{m / n}}

Fractional powers defined

Read $8^{2/3}$ as “the cube root of 8, then squared” $= 2^{2} = 4$ . Root first, then power — this keeps the numbers small and the work easy.

Example 4 · Basic

Find

2 7^{2/3}

and

1 6^{- 3/4}

.

Solution, step by step

1 $2 7^{2/3} = (327)^{2} = 3^{2} = 9$ .
2 $1 6^{- 3/4} = \frac{1}{( 4 16 ) ^{3}} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .

Answer $9$ and $\frac{1}{8}$

Example 5 · Standard

Find

(\frac{8}{27})^{- 2/3}

.

Solution, step by step

1A negative power flips the fraction: $(\frac{27}{8})^{2/3}$ .
2Cube root: $\frac{3}{2}$ ; then square: $\frac{9}{4}$ .

Answer $\frac{9}{4}$

BasicTry it yourself first

Find the value of

6 4^{\frac{1}{3}}

.

BasicTry it yourself first

Find the value of

2 5^{\frac{3}{2}}

.

StandardTry it yourself first

Find the value of

3 2^{\frac{3}{5}}

.

StandardTry it yourself first

Find the value of

8 1^{- \frac{3}{4}}

.

StandardTry it yourself first

Find the value of

12 5^{- \frac{2}{3}}

.

AdvancedTry it yourself first

Find the value of

24 3^{\frac{4}{5}}

.

AdvancedTry it yourself first

Find the value of

102 4^{- \frac{3}{10}}

.

Exponent equations

Make the bases equal

When $x$ sits in the power — make both sides the same base and equate the powers.

a^{x} = a^{y} ⟹ x = y (a > 0, a \neq = 1)

The core rule

Write both sides as powers of one base (break numbers into prime factors).
With equal bases, equate the powers.
Now solve the ordinary linear equation.

Example 6 · Basic

Find

x

in

3^{x + 2} = 243

.

Solution, step by step

1 $243 = 3^{5}$ , so $3^{x + 2} = 3^{5}$ .
2 $x + 2 = 5$ ⟹ $x = 3$

Answer 3

Example 7 · Standard

Find

x

in

5^{x} + 5^{x + 1} = 750

.

Solution, step by step

1 $5^{x} (1 + 5) = 750$ ⟹ $5^{x} = 125$ .
2 $125 = 5^{3}$ ⟹ $x = 3$

Answer 3

Example 8 · Advanced

Find

x

in

2^{x} \times 4^{x + 1} = 128

.

Solution, step by step

1 $4^{x + 1} = 2^{2 x + 2}$ , so the left side is $2^{x + 2 x + 2} = 2^{3 x + 2}$ .
2 $128 = 2^{7}$ ⟹ $3 x + 2 = 7$ ⟹ $x = \frac{5}{3}$

Answer $\frac{5}{3}$

BasicTry it yourself first

If

2^{x + 3} = 128

, what is

x

?

BasicTry it yourself first

If

5^{x - 1} = 25

, what is

x

?

StandardTry it yourself first

If

3^{2 x + 1} = 243

, what is

x

?

StandardTry it yourself first

If

7^{3 x - 2} = 7

, what is

x

?

StandardTry it yourself first

If

2^{2 x - 3} = 128

, what is

x

?

AdvancedTry it yourself first

If

4^{3 x + 2} = 65, 536

, what is

x

?

AdvancedTry it yourself first

If

3^{4 x - 5} = 27

, what is

x

?

StandardTry it yourself first

If

3^{x} + 3^{x + 1} = 108

, what is

x

?

StandardTry it yourself first

If

2^{x} + 2^{x + 1} = 96

, what is

x

?

AdvancedTry it yourself first

If

6^{x} + 6^{x + 1} = 1, 512

, what is

x

?

StandardTry it yourself first

If

2^{a} = 3^{b} = 6^{c}

with

a, b, c \neq = 0

, then

\frac{1}{a} + \frac{1}{b}

equals:

2

Surds

Spotting · operations · comparing · rationalising

What a surd is, simplest form, adding and multiplying, ordering, clearing denominators and simplifying $a \pm 2 b$ .

What a surd is

A root that does not come out whole

Definition, order, pure and mixed surds, like and unlike surds.

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Add, subtract, multiply, divide

Only like terms combine

Simplify every surd first; then add or subtract the coefficients of like surds.

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Comparing surds

LCM of the orders

Arranging surds of different orders from biggest to smallest.

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Rationalisation

Clearing the root from the denominator

If the denominator has a surd, multiply by its conjugate to make it rational.

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Roots inside roots

Simplifying

a \pm 2 b

·

x + \frac{1}{x}

Two terms whose squares add to $a$ and product is $b$ ; and finding $x + \frac{1}{x}$ from $x = p + q$ .

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3

Traps and fast tricks

Slips · fast methods

The common slips and a few tricks that save time.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Equalise the base · spot the pair

A few patterns that shrink a long calculation.

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4

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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5

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Indices

Laws of indices

Fractional and negative powers

Exponent equations

Surds

What a surd is

Add, subtract, multiply, divide

Comparing surds

Rationalisation

Roots inside roots

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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