Topics
3. Logarithmic and Exponential Functions
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Standard Level

3.01 Rational exponents

Lesson
Worksheet
Practice
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

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

12

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.

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

Consider the expression m^{\frac{1}{4}} = 8^{\frac{1}{2}}.

a

Complete the following working:

\begin{aligned} m^{\frac{1}{4}} &= (m^⬚)^{\frac{1}{2}}\\ &= \sqrt{⬚}^{\frac{1}{2}} \end{aligned}

b

Hence, or otherwise, solve the equation m^{\frac{1}{4}} = 8^{\frac{1}{2}}.

16

Consider the equation x^{\frac{2}{3}} = 9.

a

Complete the statement: \left(x^{\frac{2}{3}}\right)^{⬚} = x

b

Hence, solve the equation x^{\frac{2}{3}} = 9.

17

Find the missing values in the following equations:

a
16^{\frac{3}{⬚}} = 64
b
\left(⬚\right)^{\frac{2}{3}} = 25
c
625^{\frac{3}{⬚}} = 125
d
\left(⬚\right)^{\frac{4}{3}} = 81
18

Solve the following equations for x:

a

x^{\frac{3}{4}} = 125

b

\left(x - 4\right)^{0.5} = 4

c

\left(2x - 1\right)^{\frac{2}{3}} = 9

d

\left( 4 x + 3\right)^{ - 1.5 } = 8

19

Solve the following equation for k:

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

20

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.

Applications
21

The surface area A of a cube with a volume V is given by A = 6 V^{\frac{2}{3}}.

a

Find the surface area of a cube with a volume of 8 \text{ cm}^{3}.

b

Find the volume of a cube with a surface area of 216 \text{ cm}^{2}.

22

The volume V of a sphere with surface area A is given by: V = \dfrac{A^{\frac{3}{2}}}{6 \sqrt{\pi}}

a

Find the volume of a sphere with a surface area of 50 \text{ cm}^{2}. Round you answer to the nearest cubic centimetre.

b

Find the surface area of a sphere with a volume of 125 \text{ cm}^{3}. Round you answer to the nearest square centimetre.

23

The volume V of a regular tetrahedron with edge length a is given by: V = \frac{a^3}{6\sqrt{2}}

a

Find the volume of a tetrahedron with a side length of 2 \text{ cm}.

b

Rearrange the formula to make a the subject.

c

Hence, find the side length of a tetrahedon with a volume of 72 \text{ cm}^3.

24

A company's net profit (in dollars) can be modelled by the equation: P = 50 n^{\frac{3}{4}} - 2500 Where n is the number of units sold.

a

Find its net profit be in dollars if it manages to sell 256 units.

b

Find the number of units needed to be sold to make a net profit of \$8300.

25

A company's net profit (in dollars) can be modelled by the equation: P = 180 n^{\frac{3}{5}} - 4000 Where n is the number of units sold.

a

Find its net profit be in dollars if it manages to sell 32 units.

b

Find the number of units needed to be sold to make a net profit of \$45\,000. Round to the nearest whole unit.

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