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iGCSE (2021 Edition)

1.06 Irrational numbers

Worksheet
Irrational numbers
1

State which of the following are rational or irrational:

a

\sqrt{25}

b

\sqrt{361}

c

\sqrt{33}

d

\sqrt[3]{64}

e

\sqrt[3]{47}

f

\sqrt[3]{125}

g

\sqrt[3]{1000}

h

\sqrt[3]{1729}

2

Evaluate the following, rounding your answers to two decimal places where necessary:

a

\sqrt{25}

b

\sqrt[3]{64}

c

\sqrt{33}

d

\sqrt[3]{26}

e

\sqrt{121}

f

\sqrt[3]{216}

g

\sqrt{83}

h

\sqrt[3]{52}

3

Is \sqrt{15} an integer?

4

Determine whether each of the following numbers is a surd:

a

\sqrt{2}

b

\sqrt{1}

c

\sqrt{50}

d

\sqrt[3]{27}

e

\sqrt[3]{9}

f

\sqrt[3]{8}

g

\sqrt{80}

h

\sqrt{144}

5

What is the largest square number that divides exactly into\, 108?

6

Consider\sqrt{290} and answer the following questions:

a

Is \sqrt{290} an exact value?

b

A calculator states that \sqrt{290} is 17.029386366. Is this still exact?

Simplifying surds
7

Are the following expressions written in their simplest surd form?:

a

\sqrt{17}

b

\sqrt{14}

c

\sqrt{50}

d

5 \sqrt{50}

e

7 \sqrt{125}

f

11 \sqrt{21}

g

\sqrt{63}

h

\sqrt{112}

8

Simplify the following:

a

\sqrt{180}

b

\sqrt{125}

c

3 \sqrt{54}

d

7 \sqrt{32}

e

6 \sqrt{100}.

f

\sqrt{ 25 \times 6}

g

\dfrac{1}{2} \sqrt[3]{ 8 \times 6}.

h

\sqrt[3]{ 24 \times 9}.

9

Simplify the following:

a

\dfrac{1}{\sqrt{25}}

b

\dfrac{16}{\sqrt{16}}

c

\dfrac{\sqrt{64}}{8}

d

\dfrac{36}{\sqrt[3]{216}}

10

Write the expression 3 \sqrt{5} as a single surd.

11

Consider the equation x \sqrt{8} = \sqrt{648}. Find the value of the pronumeral x.

12

Consider the surd \sqrt{54}:

a

Simplify the surd.

b

Evaluate the surd to two decimal places.

13

Simplify the following:

a

\sqrt{100}

b

\sqrt{170}

c

\sqrt{64} \times \sqrt{1}

d

\sqrt[3]{64}

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Outcomes

0606C4.1B

Perform simple operations with surds.

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