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10&10a

2.06 Fractional indices

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Worksheet
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Worksheet
Positive fractional indices
1

Write the following in surd form:

a
12^\frac{1}{2}
b
15^\frac{1}{2}
c
7^\frac{1}{3}
d
35^\frac{1}{4}
e
21^\frac{1}{3}
f
31^{\frac{1}{6}}
g
10^\frac{1}{5}
h
24^\frac{1}{2}
2

For each of the following expressions:

i

Write in surd form.

ii

Evaluate the expression.

a
36^{\frac{1}{2}}
b
64^\frac{1}{2}
c
81^{\frac{1}{2}}
d
25^\frac{1}{2}
e
49^{\frac{1}{2}}
f
100^\frac{1}{2}
g
121^\frac{1}{2}
h
256^\frac{1}{2}
3

Consider the expression x^\frac{1}{3}.

a

Write the expression in surd form.

b

Evaluate the expression when x=8.

4

Complete the following statement:

14^{\frac{1}{3}} = \sqrt[3]{⬚}

5

Write the fourth root of 18 in surd form.

6

Consider the expression \left(\sqrt[4]{64}\right)^{4}.

a

Write the expression in index form.

b

Hence evaluate the expression.

7

Evaluate:

a
1000^{\frac{1}{3}}
b
64^{\frac{1}{3}}
c
256^{\frac{1}{4}}
d
32^{\frac{1}{5}}
e
64^{\frac{1}{6}}
f
1^{\frac{1}{6}}
g
27^\frac{1}{3}
h
16^\frac{1}{4}
i
64^\frac{1}{3}
j
125^\frac{1}{3}
k
128^\frac{1}{7}
l
216^\frac{1}{3}
8

Write each of the following with a fractional index:

a
\sqrt[3]{5}
b
\sqrt[8]{23}
c
\sqrt[6]{6}
d
\sqrt[3]{10}
e
\sqrt[4]{15}
f
\sqrt[7]{18}
g
\sqrt[5]{31}
h
\sqrt[8]{57}
9

Write each of the following with a fractional index:

a
\left(\sqrt[9]{19}\right)^{7}
b
\left(\sqrt[3]{7}\right)^{8}
c
\left(\sqrt[5]{12}\right)^{6}
d
\left(\sqrt[6]{45}\right)^{5}
e
\sqrt[4]{5^3}
f
\sqrt[7]{13^4}
g
\sqrt[9]{21^5}
h
\sqrt[5]{11^{12}}
10

Write the following in surd form:

a
19^{\frac{5}{3}}
b
7^{\frac{3}{2}}
c
11^{\frac{6}{5}}
d
24^{\frac{8}{7}}
11

For each of the following expressions:

i

Write in the form \left(\sqrt[n]{a}\right)^{m}.

ii

Evaluate the expression.

a
8^{\frac{2}{3}}
b
4^{\frac{5}{2}}
c
27^{\frac{4}{3}}
d
125^{\frac{2}{3}}
e
64^{\frac{2}{3}}
f
16^{\frac{7}{4}}
g
1024^{\frac{4}{5}}
h
32^{\frac{6}{5}}
12

Explain why it is usually easier to evaluate a^\frac{m}{n} by writing it as \left(\sqrt[n]{a}\right)^m, rather than by finding \sqrt[n]{a^m}. Use a numerical example in your explanation.

Negative fractional indices
13

For each of the following expressions:

i

Write in surd form.

ii

Evaluate the expression.

a
49^{-\frac{1}{2}}
b
121^{-\frac{1}{2}}
c
81^{-\frac{1}{4}}
d
8^{-\frac{1}{3}}
e
64^{-\frac{1}{3}}
f
128^{-\frac{1}{7}}
g
125^{-\frac{1}{3}}
h
256^{-\frac{1}{4}}
14

For each of the following expressions:

i

Write in surd form.

ii

Evaluate the expression.

a
16^{-\frac{3}{2}}
b
4^{-\frac{5}{2}}
c
27^{-\frac{4}{3}}
d
216^{-\frac{2}{3}}
e
243^{-\frac{2}{5}}
f
81^{-\frac{3}{4}}
g
125^{-\frac{2}{3}}
h
512^{-\frac{5}{9}}
15

Write each of the following with a fractional index:

a
\dfrac{1}{\sqrt{7}}
b
\dfrac{1}{\sqrt[3]{5}}
c
\dfrac{1}{\sqrt[5]{9}}
d
\dfrac{1}{\sqrt[4]{17}}
e
\dfrac{1}{\sqrt[7]{3}}
f
\dfrac{1}{\sqrt[6]{12}}
g
\dfrac{1}{\sqrt[3]{23}}
h
\dfrac{1}{\sqrt[8]{40}}
16

Write each of the following with a fractional index:

a
\dfrac{1}{\left(\sqrt{6}\right)^5}
b
\dfrac{1}{\left(\sqrt[3]{10}\right)^7}
c
\dfrac{1}{\left(\sqrt[8]{12}\right)^3}
d
\dfrac{1}{\left(\sqrt[6]{17}\right)^8}
e
\dfrac{1}{\sqrt[4]{9^5}}
f
\dfrac{1}{\sqrt[8]{13^7}}
g
\dfrac{1}{\sqrt[5]{20^2}}
h
\dfrac{1}{\sqrt[6]{27^{16}}}
Index laws with fractional indices
17

Simplify:

a
7^{\frac{3}{5}}\times 7^{\frac{4}{5}}
b
5^{-\frac{2}{9}}\times 5^{\frac{4}{9}}
c
3^{\frac{3}{5}}\times 3^{-\frac{1}{5}}
d
10^{\frac{4}{7}}\times 10^{-\frac{1}{7}}
e
8^{\frac{7}{5}}\div 8^{\frac{1}{5}}
f
11^{\frac{3}{5}}\div 11^{-\frac{1}{5}}
g
4^{\frac{2}{9}}\div 4^{-\frac{3}{9}}
h
15^{-\frac{4}{7}}\div 15^{-\frac{8}{7}}
18

Evaluate:

a
8^{\frac{2}{3}}\times 8^{\frac{5}{3}}
b
27^{\frac{9}{6}}\times 27^{-\frac{5}{6}}
c
64^{\frac{7}{12}}\times 64^{-\frac{5}{12}}
d
81^{\frac{5}{4}}\times 81^{-\frac{3}{4}}
e
27^{\frac{7}{3}}\div 27^{\frac{5}{3}}
f
125^{\frac{7}{3}}\div 27^{\frac{5}{3}}
g
128^{\frac{1}{7}}\div 128^{-\frac{3}{7}}
h
243^{-\frac{3}{5}}\div 243^{-\frac{1}{5}}
19

Write \sqrt{3} \times \sqrt[3]{3} in surd form.

20

Evaluate:

a
\left(\dfrac{4}{9}\right)^{\frac{1}{3}} \times \left(\dfrac{4}{9}\right)^{\frac{1}{3}}
b
\left(\dfrac{3}{5}\right)^{\frac{3}{4}} \times \left(\dfrac{3}{5}\right)^{-\frac{7}{4}}
c
\left(\dfrac{8}{27}\right)^{\frac{4}{3}} \div \left(\dfrac{8}{27}\right)^{\frac{2}{3}}
d
\left(\dfrac{343}{216}\right)^{\frac{1}{3}} \div \left(\dfrac{343}{216}\right)^{-\frac{1}{3}}
21

Evaluate:

a

\dfrac{1000^{\frac{1}{3}}}{1000^{\frac{2}{3}}}

b

\left(\dfrac{243}{243^{\frac{1}{5}}}\right)^{\frac{1}{4}}

22

Evaluate:

a
\left(16^{\frac{1}{3}}\right)^{\frac{3}{4}}
b
\left(64^{\frac{1}{6}}\right)^{\frac{6}{3}}
c
\left(243^{\frac{2}{3}}\right)^{\frac{6}{5}}
d
\left(256^{\frac{1}{5}}\right)^{\frac{5}{8}}
e
\left(25^{\frac{1}{5}}\right)^{-\frac{5}{2}}
f
\left(81^{-\frac{1}{3}}\right)^{\frac{9}{4}}
g
\left(128^{\frac{6}{7}}\right)^{-\frac{3}{2}}
h
\left(27^{-\frac{5}{4}}\right)^{-\frac{4}{3}}
23

Evaluate:

a
\left(\dfrac{4^{4}}{81}\right)^{\frac{1}{4}}
b
\left(\dfrac{3^{6}}{64}\right)^{\frac{1}{6}}
c
\left(\dfrac{2^{3}}{27}\right)^{\frac{2}{3}}
d
\left(\dfrac{27^{2}}{64}\right)^{\frac{5}{6}}
e
\left(\dfrac{5^4}{256}\right)^{-\frac{1}{4}}
f
\left(\dfrac{11^5}{243}\right)^{-\frac{1}{5}}
g
\left(\dfrac{16}{5^4}\right)^{-\frac{3}{4}}
h
\left(\dfrac{2^8}{81}\right)^{-\frac{5}{4}}
24

Evaluate:

a
\left(\dfrac{8}{27}\right)^{\frac{1}{3}}
b
\left(\dfrac{81}{16}\right)^{\frac{5}{4}}
c
\left(\dfrac{64}{125}\right)^{\frac{2}{3}}
d
\left(\dfrac{625}{16}\right)^{\frac{1}{4}}
e
\left(\dfrac{64}{27}\right)^{\frac{2}{3}}
f
\left(\dfrac{125}{216}\right)^{\frac{2}{3}}
g
\left(\dfrac{27}{8}\right)^{\frac{4}{3}}
h
\left(\dfrac{1000}{27}\right)^{\frac{4}{3}}
25

Evaluate:

a
\left(\dfrac{16}{9}\right)^{-\frac{1}{2} }
b
\left(\dfrac{25}{64}\right)^{-\frac{1}{2} }
c
\left(\dfrac{27}{8}\right)^{-\frac{4}{3} }
d
\left(\dfrac{16}{81}\right)^{-\frac{3}{4} }
e
\left(\dfrac{64}{125}\right)^{-\frac{2}{3} }
f
\left(\dfrac{16}{625}\right)^{-\frac{3}{4} }
g
\left(\dfrac{64}{27}\right)^{-\frac{2}{3} }
h
\left(\dfrac{729}{343}\right)^{-\frac{2}{3} }
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Outcomes

VCMNA355 (10a)

Define rational and irrational numbers and perform operations with surds and fractional indices.

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