topic badge

7.02 Fractional indices

Worksheet
Fractional indices
1

Write the following in surd form:

a
x^{\frac{1}{3}}
b
a^{\frac{3}{2} }
c
20^{\frac{5}{4}}
d
y^{ - \frac{1}{2} }
e
3x^{\frac{1}{2}}
f
b^{2\frac{1}{3} }
g
y^{1.5}
h
5^{ - \frac{4}{3} }
2

Write the following as a single power of 2:

a
\sqrt[3]{2}
b
\sqrt[5]{16}
c
\sqrt{32}
d
2\sqrt{2}
e
8\sqrt[3]{2}
f
2^a \times\sqrt{2}
g
\dfrac{8}{\sqrt{2}}
h
\dfrac{1}{\sqrt{8}}
3

Write the following as a single power of 3:

a
\sqrt{27}
b
3^x\times \sqrt[4]{3}
c
9\sqrt{3}
d
\dfrac{1}{\sqrt[3]{9^{x}}}
4

Complete the following statement: \begin{aligned} \sqrt{m^{8}} &=\left(m^{8}\right)^{⬚}\\ &=m^{ ⬚ \times \frac{1}{2}}\\ &=m^{⬚} \end{aligned}

5

Write the following in the form x^k, where k is rational:

a

\sqrt{x}

b

\sqrt[6]{x}

c

\dfrac{1}{\sqrt{x}}

d
\left(\sqrt[3]{x}\right)^{2}
e
\sqrt[4]{x^3}
f
\left(\sqrt[7]{x}\right)^{6}
g
\sqrt[4]{x^8}
h
\dfrac{1}{\sqrt{x}}
i
\sqrt{\sqrt{x}}
j
\sqrt[3]{x^{ 6 a}}
k
\dfrac{1}{\sqrt{x^{ - 3 }}}
l
x\sqrt{x}
m
\dfrac{1}{\sqrt[4]{x^{5}}}
n
\sqrt[5]{\sqrt{x}}
o
x^2\times \sqrt[3]{x}
6

Consider the expression 64^{\frac{2}{3}}.

a

Complete the statement: 64^{\frac{2}{3}} = \left(\sqrt[⬚]{64}\right)^⬚

b

Hence, evaluate 64^{\frac{2}{3}}.

7

Without using a calculator, evaluate the following:

a
1^{\frac{1}{10}}
b
4^{\frac{3}{2}}
c
121^{\frac{1}{2}}
d
\sqrt[3]{27}
e
125^{\frac{1}{3}}
f
\left( - 64 \right)^{\frac{1}{3}}
g
64^{ - \frac{1}{6} }
h
81^{ - \frac{3}{4} }
i
\sqrt[3]{ - 64 }
j
\left(\dfrac{9}{100}\right)^{\frac{1}{2}}
k
\left(64^{\frac{1}{9}}\right)^{\frac{9}{2}}
l
\left(\dfrac{8}{125}\right)^{\frac{2}{3}}
m
\left( - 32 \right)^{\frac{4}{5}}
n
\dfrac{64^{\frac{1}{3}}}{64^{\frac{2}{3}}}
o
\dfrac{1}{- \sqrt{169}}
p
\sqrt[3]{\dfrac{- 64}{125}}
q
\dfrac{\sqrt[3]{40}}{\sqrt[3]{5}}
r
36^{\frac{1}{2}} - 32^{\frac{3}{5}}
s
1000^{\frac{8}{9}} \times 1000^{ -\frac{5}{9} }
t
\sqrt[3]{\sqrt[4]{16} + \sqrt{625}}
8

Use a calculator to evaluate the following, to two decimal places:

a
10^{\frac{3}{2}}
b
4^{\frac{5}{3}}
c
\sqrt[4]{5^3}
d
\left(\sqrt[3]{12}\right)^5
9

Simplify the following expressions, giving your answers in index form. Assume that all variables represent positive numbers.

a
\left(\sqrt{b}\right)^8
b
\sqrt{m^{6}}
c
\sqrt[4]{a^{5}}
d
\dfrac{1}{\sqrt[5]{a^{6}}}
e
\sqrt[3]{x^{6}}
f
\sqrt[3]{m^{3}}
g
\sqrt{ j^{2} x^{4}}
h
\left(\sqrt[4]{ x^{5} y^{3}}\right)^{24}
i
\left( 4 a^{8}\right)^{\frac{1}{2}}
j
\left( 625 u^{16} v^{12}\right)^{\frac{1}{2}}
k
8 b^{\frac{3}{4}} \div 2 b^{\frac{2}{3}}
l
y^{3} \times \sqrt[3]{y}
m
\sqrt{\left( 2 x + 9\right)^{2}}
n
\sqrt[3]{ 9^{3} x^{18} y^{12}}
o
\left(\dfrac{x^{15}}{1024}\right)^{\frac{2}{5}}
p
\sqrt{\dfrac{36 x^{18}}{y^{20}}}
q
\sqrt{ 16 y^{2} + 24 y + 9}
r
\sqrt{ x^{2} y^{2} + 18 x y^{2} + 81 y^{2}}
10

Simplify the following expressions, giving your answers in surd form. Assume that all variables represent positive numbers.

a
\dfrac{\sqrt{x^{2} + 5 x + 6}}{\sqrt{x + 2}}
b
\dfrac{\sqrt[3]{x^{2} + 9 x + 20}}{\sqrt[3]{x + 4}}
11

Determine whether the following statements accurately describe the meaning of the expression x^{ - \frac{y}{z} }:

a

x^{ - \frac{y}{z} } means we are raising x to the power of \dfrac{z}{y}, then taking the reciprocal of the result.

b

x^{ - \frac{y}{z} } means we are taking the reciprocal of x, then raising the result to the power of \dfrac{y}{z}.

c

x^{ - \frac{y}{z} } means we are taking the reciprocal of x, then raising the result to the power of \dfrac{z}{y}.

d

x^{ - \frac{y}{z} } means we are raising x to the power of \dfrac{y}{z}, then taking the reciprocal of the result.

12

Describe how we could interpret the expression m^{\frac{q}{r}} in terms of powers and roots of m.

13

Is there a real number that equals \sqrt[4]{ - 16 }? Explain your answer.

14

Consider the expression m^{5} \times m \sqrt{m}.

a

Express it in simplest index form.

b

Express it in surd form.

15

Solve the following equation for k:

\sqrt[k]{y} \times \sqrt[k]{y} \times \sqrt[k]{y} = y^{\frac{1}{2}}

16

To evaluate 81^{\frac{3}{2}} would it be more efficient to use the property a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}, or the property a^{\frac{m}{n}} = \sqrt[n]{a^{m}}. Explain your answer.

17

Simplify the following, giving your answers with positive indices. Assume that all variables represent positive numbers.

a

\dfrac{2}{m^{5}} \times \dfrac{m^{6}}{5}

b

\dfrac{4}{m^{4}} \times \dfrac{m^{2}}{5}

c

\dfrac{3}{m^{\frac{5}{9}}} \times \dfrac{m^{\frac{7}{9}}}{4}

d

\dfrac{2}{m^{\frac{7}{9}}} \times \dfrac{m^{\frac{3}{9}}}{3}

e

\dfrac{3}{\sqrt{m^{7}}} \times \dfrac{\sqrt{m^{3}}}{4}

f

\dfrac{\sqrt[3]{m^{5}}}{4} \times \dfrac{3}{\sqrt[3]{m^{7}}}

g

\dfrac{\sqrt{m^{7}}}{\sqrt[4]{m^{5}}} \times \dfrac{\sqrt{m^{9}}}{\sqrt[4]{m^{3}}}

h

\dfrac{3}{m^{\frac{2}{7}}} \div \dfrac{4}{m^{\frac{5}{7}}}

i

\dfrac{m^{\frac{7}{3}}}{m^{\frac{3}{2}}} \div \dfrac{m^{\frac{1}{2}}}{m^{\frac{8}{3}}}

j

\dfrac{m^{\frac{1}{3}}}{m^{\frac{11}{2}}} \div \dfrac{m^{\frac{7}{2}}}{m^{\frac{8}{3}}}

k

\dfrac{m^{5}}{m^{3}} \div \dfrac{m^{2}}{m^{7}}

l

\dfrac{m^{2}}{m^{5}} \div \dfrac{m^{4}}{m^{3}}

m

\dfrac{x^{\frac{6}{7}} + x^{\frac{5}{7}}}{x^{\frac{4}{7}}}

n

\dfrac{x^{\frac{3}{5}} + x^{\frac{2}{5}}}{x^{\frac{4}{5}}}

o

\dfrac{x^{\frac{8}{9}} - x^{\frac{7}{9}}}{x^{\frac{2}{9}} \times x^{\frac{3}{9}}}

p

\dfrac{x^{\frac{8}{9}} - x^{\frac{5}{9}}}{x^{\frac{4}{9}} \times x^{\frac{3}{9}}}

18

Simplify the following, giving your answers in surd form with positive indices. Assume that all variables represent positive numbers.

a

\dfrac{2}{\sqrt{m^{4}}} \div \dfrac{\sqrt{m^{7}}}{5}

b

\dfrac{\sqrt[3]{m^{2}}}{2} \div \dfrac{5}{\sqrt[3]{m^{5}}}

19

Simplify the following:

a
\dfrac{5^{x} + 5^{\frac{x}{2}} - 6}{5^{\frac{x}{2}} - 2}
b
\dfrac{2^{x} + 2^{\frac{x}{2} + 2} + 4}{2^{\frac{x}{2}} + 2}
Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

2.1.1

review indices (including fractional and negative indices) and the index laws

2.1.2

use radicals and convert to and from fractional indices

What is Mathspace

About Mathspace