Indices and Surds — Extra Practice

Competitive Mathematics · Number System

5 Part19 worked examples151 practice questions

NoteThis is the Extra Practice set: lots of questions. To learn the methods first, read the main chapter of the same topic (the one without “Extra Practice” in its name), then practise here.

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).

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 $a^{- n}$ .

∑ Formula

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.

∑ Formula

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

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

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

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

2^{x + 3} = 128

, what is

x

BasicTry it yourself first

5^{x - 1} = 25

, what is

x

StandardTry it yourself first

3^{2 x + 1} = 243

, what is

x

StandardTry it yourself first

7^{3 x - 2} = 7

, what is

x

StandardTry it yourself first

2^{2 x - 3} = 128

, what is

x

AdvancedTry it yourself first

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

, what is

x

AdvancedTry it yourself first

3^{4 x - 5} = 27

, what is

x

StandardTry it yourself first

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

, what is

x

StandardTry it yourself first

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

, what is

x

AdvancedTry it yourself first

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

, what is

x

StandardTry it yourself first

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

with

a, b, c \neq = 0

, then

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

equals:

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.

A root of a rational number that is not itself rational is a surd: $2$ , $35$ , $7$ are surds; $16 = 4$ is not because it comes out whole. Every surd is irrational, but not every irrational number is a surd ( $π$ is not).

Term	Meaning	Example
Order	Which root — square, cube, fourth…	$35$ → 3
Radicand	The number under the root sign	$7$ → 7
Quadratic surd	Order 2	$3$
Pure surd	No rational coefficient	$18$
Mixed surd	Rational × surd	$32$
Like surds	Same radicand and order	$25, 75$
Unlike surds	Different radicand	$2, 3$

∑ Formula

n a \times n b = n ab, \frac{n a}{n b} = n \frac{a}{b}, m n a = mn a

Rules of surds

Example 9 · Basic

Write

72

and

354

in simplest form.

Solution, step by step

1 $72 = 36 \times 2$ , $36 = 6$ ⟹ $62$ .
2 $54 = 27 \times 2$ , $327 = 3$ ⟹ $332$ .

Answer $62$ , $332$

BasicTry it yourself first

Write

45

in simplest form.

BasicTry it yourself first

Write

98

in simplest form.

StandardTry it yourself first

Write

288

in simplest form.

StandardTry it yourself first

Write

1620

in simplest form.

StandardTry it yourself first

Write

3250

in simplest form.

AdvancedTry it yourself first

Write

31458

in simplest form.

Add, subtract, multiply, divide

Only like terms combine

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

Add / subtract: bring each surd to simplest form and combine the coefficients of the same radicand. $50 + 18 = 52 + 32 = 82$ . $2 + 3$ cannot be made one term — they are unlike.

Multiply / divide: with the same order multiply or divide the radicands; with different orders first bring them to a common order using the LCM. $2 \times 32 = 2^{1/2} \times 2^{1/3} = 2^{5/6} = 632$ .

∑ Formula

(a + b) (a - b) = a - b, (a \pm b)^{2} = a + b \pm 2 ab

The two most useful identities

Example 10 · Standard

Evaluate

48 + 75 - 27

Solution, step by step

1 $48 = 43$ , $75 = 53$ , $27 = 33$ .
2 $(4 + 5 - 3) 3 = 63$

Answer $63$

Example 11 · Standard

Evaluate

(3 + 5) (3 - 5)

and

(7 + 2)^{2}

Solution, step by step

1First: $3^{2} - (5)^{2} = 9 - 5 = 4$ .
2Second: $7 + 2 + 214 = 9 + 214$ .

Answer $4$ and $9 + 214$

Example 12 · Advanced

Which is bigger:

2 \times 33

34

Solution, step by step

1LCM of orders 2 and 3 is 6: $2 = 6 2^{3} = 68$ , $33 = 69$ .
2The product is $672$ and $34 = 616$ ; $72 > 16$ .

Answer $2 \times 33$

StandardTry it yourself first

What is the value of

48 + 75 - 27

StandardTry it yourself first

What is the value of

50 + 18

StandardTry it yourself first

What is the value of

180 - 20 + 45

AdvancedTry it yourself first

What is the value of

567 - 112 + 28

BasicTry it yourself first

What is

(11 + 3) (11 - 3)

StandardTry it yourself first

What is

(5 + 3)^{2}

Comparing surds

LCM of the orders

Arranging surds of different orders from biggest to smallest.

Find the LCM of the orders of all the surds.
Convert each to that common order: $m a = L a^{L / m}$ .
Now compare the radicands — the bigger radicand gives the bigger surd.

Example 13 · Standard

Arrange

3

34

45

from biggest to smallest.

Solution, step by step

1LCM of 2, 3, 4 is 12.
2 $3 = 12 3^{6} = 12729$ ; $34 = 12 4^{4} = 12256$ ; $45 = 12 5^{3} = 12125$ .
3Order: $3 > 34 > 45$

Answer $3 > 34 > 45$

BasicTry it yourself first

Which of these is the greatest?

2

33

65

StandardTry it yourself first

Which of these is the greatest?

7

311

430

StandardTry it yourself first

Which of these is the smallest?

32

43

65

AdvancedTry it yourself first

Which of these is the greatest?

6

312

420

6100

AdvancedTry it yourself first

Which of these is the smallest?

39

420

660

5

Rationalisation

Clearing the root from the denominator

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

A surd in the denominator makes the value hard to find. So we multiply top and bottom by an expression that makes the denominator rational — the rationalising factor.

Denominator	Multiply by	New denominator
$a$	$a$	$a$
$a + b$	$a - b$	$a - b$
$p + b$	$p - b$	$p^{2} - b$
$3 a$	$3 a^{2}$	$a$

Example 14 · Basic

Simplify

\frac{10}{5}

Solution, step by step

1Multiply top and bottom by $5$ : $\frac{10 5}{5}$ .
2 $= 25$

Answer $25$

Example 15 · Standard

Rationalise

\frac{1}{3 + 8}

Solution, step by step

1Multiply by the conjugate $3 - 8$ : denominator $= 9 - 8 = 1$ .
2Value $= 3 - 8 = 3 - 22$

Answer $3 - 22$

Example 16 · Advanced

3 = 1.732

, what is

\frac{4}{3 - 1}

Solution, step by step

1Multiply by the conjugate $3 + 1$ : $\frac{4 ( 3 + 1 )}{3 - 1} = 2 (3 + 1)$ .
2 $= 2 \times 2.732 = 5.464$

Answer 5.464

BasicTry it yourself first

Simplify

\frac{21}{7}

StandardTry it yourself first

Simplify

\frac{30}{6}

StandardTry it yourself first

Simplify

\frac{44}{11}

StandardTry it yourself first

Rationalise and simplify

\frac{3}{5 + 2}

StandardTry it yourself first

Rationalise and simplify

\frac{8}{7 + 3}

AdvancedTry it yourself first

Rationalise and simplify

\frac{27}{11 + 2}

AdvancedTry it yourself first

Rationalise and simplify

\frac{18}{13 + 4}

StandardTry it yourself first

What is the value of

\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{9 + 10}

AdvancedTry it yourself first

What is the value of

\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{25 + 26}

StandardTry it yourself first

3 = 1.732

, what is the approximate value of

\frac{6}{3}

StandardTry it yourself first

5 = 2.236

, what is the approximate value of

\frac{10}{5}

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 $x = p + q$ .

To write $a + 2 b$ as $x + y$ find $x, y$ with $x + y = a$ and $x y = b$ . Then $(x + y)^{2} = x + y + 2 x y = a + 2 b$ . For the minus form, $a - 2 b = x - y$ (with $x > y$ ).

Example 17 · Standard

Simplify

7 + 43

Solution, step by step

1 $43 = 212$ , so $a = 7$ , $b = 12$ . We need $x, y$ with sum 7 and product 12: $x = 4, y = 3$ .
2 $7 + 43 = 4 + 3 = 2 + 3$

Answer $2 + 3$

Example 18 · Standard

x = 3 + 22

, find

x + \frac{1}{x}

and

x^{2} + \frac{1}{x ^{2}}

Solution, step by step

1 $\frac{1}{x} = \frac{3 - 2 2}{9 - 8} = 3 - 22$ .
2 $x + \frac{1}{x} = 6$ and $x^{2} + \frac{1}{x ^{2}} = 6^{2} - 2 = 34$ .

Answer $6$ and $34$

BasicTry it yourself first

Simplify

7 + 43

StandardTry it yourself first

Simplify

11 + 62

StandardTry it yourself first

Simplify

21 + 85

AdvancedTry it yourself first

Simplify

31 + 106

StandardTry it yourself first

x = 2 + 3

, what is

x + \frac{1}{x}

StandardTry it yourself first

x = 3 + 8

, what is

x^{2} + \frac{1}{x ^{2}}

AdvancedTry it yourself first

x = 4 + 15

, what is

x^{3} + \frac{1}{x ^{3}}

AdvancedTry it yourself first

x = 5 + 24

, what is

x^{2} + \frac{1}{x ^{2}}

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.

Trap 1Adding powers when the bases differ.

2^{3} \times 3^{3} \neq = 6^{6}

; it is

(2 \times 3)^{3} = 6^{3}

. The rules need the same base (or the same index).

Trap 2Treating $(a + b)^{n}$ as $a^{n} + b^{n}$ .

(a + b)^{2} = a^{2} + 2 ab + b^{2}

. A power does not spread over a sum.

Trap 3Reading a negative index as a negative number.

2^{- 3} = \frac{1}{8}

, not

- 8

Trap 4Writing $a + b = a + b$ .

9 + 16 = 5

but

9 + 16 = 7

Trap 5Adding unlike surds.

2 + 3 \neq = 5

. Simplify first, then check the radicands match.

Trap 6Forgetting to flip only the sign of the surd. The conjugate of

(a + b)

(a - b)

, not

(- a + b)

Fast tricks

Equalise the base · spot the pair

A few patterns that shrink a long calculation.

2, 4, 8, 16, 32, 64 — write all in base 2; 3, 9, 27, 81 — base 3; 5, 25, 125 — base 5.
$a^{x} + a^{x + 1} = a^{x} (1 + a)$ — taking the common power out works in every such question.
$a \times a = a$ , so $(a)^{4} = a^{2}$ ; the cube of $3 a$ is $a$ .
In $x = p + q$ , if $p^{2} - q = 1$ then $\frac{1}{x} = p - q$ and $x + \frac{1}{x} = 2 p$ — with no division.
$\frac{1}{k + k + 1} = k + 1 - k$ — such a series telescopes to $n - m$ .

Example 19 · Advanced

2 = 1.414

and

3 = 1.732

, what is

\frac{1}{3 + 2}

Solution, step by step

1By the conjugate: $\frac{3 - 2}{3 - 2} = 3 - 2$ .
2 $= 1.732 - 1.414 = 0.318$

Answer 0.318

StandardTry it yourself first

What is the value of

(2^{- 1} + 3^{- 1})^{- 1}

AdvancedTry it yourself first

What is the value of

(4^{- 1} + 6^{- 1})^{- 1}

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

BasicTry it yourself first

What is the value of

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

BasicTry it yourself first

What is the value of

\frac{1 0 ^{5} \times 1 0 ^{4}}{1 0 ^{6}}

BasicTry it yourself first

What is the value of

\frac{6 ^{9} \times 6 ^{2}}{6 ^{8}}

BasicTry it yourself first

What is

(3^{2})^{3}

BasicTry it yourself first

What is

(2^{3})^{3}

BasicTry it yourself first

Find the value of

3 6^{\frac{1}{2}}

BasicTry it yourself first

Find the value of

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

BasicTry it yourself first

Find the value of

10 0^{- \frac{1}{2}}

BasicTry it yourself first

2^{x + 2} = 32

, what is

x

BasicTry it yourself first

3^{x} = 81

, what is

x

BasicTry it yourself first

1 0^{x - 1} = 100

, what is

x

BasicTry it yourself first

Write

50

in simplest form.

BasicTry it yourself first

Write

63

in simplest form.

BasicTry it yourself first

Write

108

in simplest form.

BasicTry it yourself first

Simplify

\frac{12}{3}

BasicTry it yourself first

Simplify

\frac{12}{2}

BasicTry it yourself first

What is the value of

18 + 32

BasicTry it yourself first

What is the value of

75 - 12

BasicTry it yourself first

What is

2 \times 8 \times 9

BasicTry it yourself first

Which of these is not a surd?

BasicTry it yourself first

Which of these is the greatest?

3

35

67

Level 2 — Standard

Two ideas combined

Exam-level questions.

StandardTry it yourself first

What is the value of

\frac{3 ^{9} \times 3 ^{7}}{3 ^{13}}

StandardTry it yourself first

What is the value of

\frac{5 ^{3} \times 5 ^{4}}{5 ^{9}}

StandardTry it yourself first

What is the value of

\frac{4 ^{6} \times 8 ^{2}}{1 6 ^{5}}

StandardTry it yourself first

What is the value of

\frac{4 ^{3} \times 8 ^{4}}{1 6 ^{7}}

StandardTry it yourself first

What is the value of

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

StandardTry it yourself first

What is the value of

\frac{7 ^{n + 2} - 7 ^{n}}{7 ^{n - 1}}

StandardTry it yourself first

What is the value of

(3^{- 1} + 5^{- 1})^{- 1}

StandardTry it yourself first

Find the value of

21 6^{\frac{2}{3}}

StandardTry it yourself first

Find the value of

1 6^{- \frac{5}{4}}

StandardTry it yourself first

Find the value of

6 4^{\frac{5}{6}}

StandardTry it yourself first

5^{2 x - 1} = 3, 125

, what is

x

StandardTry it yourself first

2^{3 x + 2} = 16, 384

, what is

x

StandardTry it yourself first

3^{2 x - 2} = 81

, what is

x

StandardTry it yourself first

4^{x} + 4^{x + 1} = 80

, what is

x

StandardTry it yourself first

7^{x} + 7^{x + 1} = 392

, what is

x

StandardTry it yourself first

Write

405

in simplest form.

StandardTry it yourself first

Write

700

in simplest form.

StandardTry it yourself first

Write

3686

in simplest form.

StandardTry it yourself first

Rationalise and simplify

\frac{4}{6 + 2}

StandardTry it yourself first

Rationalise and simplify

\frac{21}{10 + 3}

StandardTry it yourself first

Simplify

\frac{26}{13}

StandardTry it yourself first

What is the value of

98 + 18 - 8

StandardTry it yourself first

What is the value of

96 + 54 + 6

StandardTry it yourself first

What is

(7 + 5)^{2} + (7 - 5)^{2}

StandardTry it yourself first

What is

(11 + 3)^{2} + (11 - 3)^{2}

StandardTry it yourself first

What is

6 \times 10 \times 15

StandardTry it yourself first

Simplify

43 + 127

StandardTry it yourself first

Simplify

27 + 102

StandardTry it yourself first

x = 9 + 80

, what is

x + \frac{1}{x}

StandardTry it yourself first

x = 7 + 48

, what is

x^{2} + \frac{1}{x ^{2}}

StandardTry it yourself first

Which of these is the smallest?

5

49

620

StandardTry it yourself first

Which of these is the greatest?

3

34

510

StandardTry it yourself first

What is the value of

\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{16 + 17}

StandardTry it yourself first

2 = 1.414

, what is the approximate value of

\frac{14}{2}

Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

AdvancedTry it yourself first

What is the value of

\frac{2 ^{13} \times 2 ^{11}}{2 ^{27}}

AdvancedTry it yourself first

What is the value of

\frac{4 ^{9} \times 8 ^{7}}{1 6 ^{12}}

AdvancedTry it yourself first

What is the value of

\frac{5 ^{n + 3} - 5 ^{n}}{5 ^{n - 2}}

AdvancedTry it yourself first

What is the value of

\frac{2 ^{n + 6} - 2 ^{n}}{2 ^{n - 4}}

AdvancedTry it yourself first

What is the value of

(6^{- 1} + 9^{- 1})^{- 1}

AdvancedTry it yourself first

Find the value of

51 2^{\frac{7}{9}}

AdvancedTry it yourself first

Find the value of

12 8^{- \frac{5}{7}}

AdvancedTry it yourself first

Find the value of

72 9^{\frac{5}{6}}

AdvancedTry it yourself first

2^{5 x - 3} = 1, 31, 072

, what is

x

AdvancedTry it yourself first

3^{3 x - 4} = 243

, what is

x

AdvancedTry it yourself first

5^{2 x + 7} = 4, 88, 28, 125

, what is

x

AdvancedTry it yourself first

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

, what is

x

AdvancedTry it yourself first

3^{x} + 3^{x + 1} = 2, 916

, what is

x

AdvancedTry it yourself first

Write

3920

in simplest form.

AdvancedTry it yourself first

Write

319683

in simplest form.

AdvancedTry it yourself first

Rationalise and simplify

\frac{48}{17 + 5}

AdvancedTry it yourself first

Rationalise and simplify

\frac{125}{29 + 4}

AdvancedTry it yourself first

What is the value of

432 - 147 + 75

AdvancedTry it yourself first

What is the value of

605 + 180 - 80

AdvancedTry it yourself first

What is

(23 + 17)^{2} + (23 - 17)^{2}

AdvancedTry it yourself first

What is

21 \times 35 \times 15

AdvancedTry it yourself first

Simplify

59 + 1410

AdvancedTry it yourself first

Simplify

77 + 1613

AdvancedTry it yourself first

x = 13 + 168

, what is

x + \frac{1}{x}

AdvancedTry it yourself first

x = 6 + 35

, what is

x^{3} + \frac{1}{x ^{3}}

AdvancedTry it yourself first

Which of these is the greatest?

311

430

670

7

AdvancedTry it yourself first

Which of these is the smallest?

4

39

410

630

AdvancedTry it yourself first

What is the value of

\frac{1}{1 + 2} + \frac{1}{2 + 3} + \dots + \frac{1}{49 + 50}

AdvancedTry it yourself first

5 = 2.236

, what is the approximate value of

\frac{25}{5}

AdvancedTry it yourself first

x = 3 + 2

, what is

x^{2} + \frac{1}{x ^{2}}

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.