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Topics

2 Surds

2.01 Irrational numbers

2.02 Adding and subtracting surds

2.03 Multiplying and dividing surds

2.04 Binomial expansions with surds

2.05 Rationalising the denominator

2.06 Fractional indices

10&10a

2.06 Fractional indices

Lesson

Worksheet

Practice

Worksheet

Positive fractional indices

Write the following in surd form:

12^\frac{1}{2}

15^\frac{1}{2}

7^\frac{1}{3}

35^\frac{1}{4}

21^\frac{1}{3}

31^{\frac{1}{6}}

10^\frac{1}{5}

24^\frac{1}{2}

For each of the following expressions:

Write in surd form.

Evaluate the expression.

36^{\frac{1}{2}}

64^\frac{1}{2}

81^{\frac{1}{2}}

25^\frac{1}{2}

49^{\frac{1}{2}}

100^\frac{1}{2}

121^\frac{1}{2}

256^\frac{1}{2}

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

Write the expression in surd form.

Evaluate the expression when x=8.

Complete the following statement:

14^{\frac{1}{3}} = \sqrt[3]{⬚}

Write the fourth root of 18 in surd form.

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

Write the expression in index form.

Hence evaluate the expression.

Evaluate:

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

64^{\frac{1}{3}}

256^{\frac{1}{4}}

32^{\frac{1}{5}}

64^{\frac{1}{6}}

1^{\frac{1}{6}}

27^\frac{1}{3}

16^\frac{1}{4}

64^\frac{1}{3}

125^\frac{1}{3}

128^\frac{1}{7}

216^\frac{1}{3}

Write each of the following with a fractional index:

\sqrt[3]{5}

\sqrt[8]{23}

\sqrt[6]{6}

\sqrt[3]{10}

\sqrt[4]{15}

\sqrt[7]{18}

\sqrt[5]{31}

\sqrt[8]{57}

Write each of the following with a fractional index:

\left(\sqrt[9]{19}\right)^{7}

\left(\sqrt[3]{7}\right)^{8}

\left(\sqrt[5]{12}\right)^{6}

\left(\sqrt[6]{45}\right)^{5}

\sqrt[4]{5^3}

\sqrt[7]{13^4}

\sqrt[9]{21^5}

\sqrt[5]{11^{12}}

Write the following in surd form:

19^{\frac{5}{3}}

7^{\frac{3}{2}}

11^{\frac{6}{5}}

24^{\frac{8}{7}}

For each of the following expressions:

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

Evaluate the expression.

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

4^{\frac{5}{2}}

27^{\frac{4}{3}}

125^{\frac{2}{3}}

64^{\frac{2}{3}}

16^{\frac{7}{4}}

1024^{\frac{4}{5}}

32^{\frac{6}{5}}

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

For each of the following expressions:

Write in surd form.

Evaluate the expression.

49^{-\frac{1}{2}}

121^{-\frac{1}{2}}

81^{-\frac{1}{4}}

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

64^{-\frac{1}{3}}

128^{-\frac{1}{7}}

125^{-\frac{1}{3}}

256^{-\frac{1}{4}}

For each of the following expressions:

Write in surd form.

Evaluate the expression.

16^{-\frac{3}{2}}

4^{-\frac{5}{2}}

27^{-\frac{4}{3}}

216^{-\frac{2}{3}}

243^{-\frac{2}{5}}

81^{-\frac{3}{4}}

125^{-\frac{2}{3}}

512^{-\frac{5}{9}}

Write each of the following with a fractional index:

\dfrac{1}{\sqrt{7}}

\dfrac{1}{\sqrt[3]{5}}

\dfrac{1}{\sqrt[5]{9}}

\dfrac{1}{\sqrt[4]{17}}

\dfrac{1}{\sqrt[7]{3}}

\dfrac{1}{\sqrt[6]{12}}

\dfrac{1}{\sqrt[3]{23}}

\dfrac{1}{\sqrt[8]{40}}

Write each of the following with a fractional index:

\dfrac{1}{\left(\sqrt{6}\right)^5}

\dfrac{1}{\left(\sqrt[3]{10}\right)^7}

\dfrac{1}{\left(\sqrt[8]{12}\right)^3}

\dfrac{1}{\left(\sqrt[6]{17}\right)^8}

\dfrac{1}{\sqrt[4]{9^5}}

\dfrac{1}{\sqrt[8]{13^7}}

\dfrac{1}{\sqrt[5]{20^2}}

\dfrac{1}{\sqrt[6]{27^{16}}}

Index laws with fractional indices

Simplify:

7^{\frac{3}{5}}\times 7^{\frac{4}{5}}

5^{-\frac{2}{9}}\times 5^{\frac{4}{9}}

3^{\frac{3}{5}}\times 3^{-\frac{1}{5}}

10^{\frac{4}{7}}\times 10^{-\frac{1}{7}}

8^{\frac{7}{5}}\div 8^{\frac{1}{5}}

11^{\frac{3}{5}}\div 11^{-\frac{1}{5}}

4^{\frac{2}{9}}\div 4^{-\frac{3}{9}}

15^{-\frac{4}{7}}\div 15^{-\frac{8}{7}}

Evaluate:

8^{\frac{2}{3}}\times 8^{\frac{5}{3}}

27^{\frac{9}{6}}\times 27^{-\frac{5}{6}}

64^{\frac{7}{12}}\times 64^{-\frac{5}{12}}

81^{\frac{5}{4}}\times 81^{-\frac{3}{4}}

27^{\frac{7}{3}}\div 27^{\frac{5}{3}}

125^{\frac{7}{3}}\div 27^{\frac{5}{3}}

128^{\frac{1}{7}}\div 128^{-\frac{3}{7}}

243^{-\frac{3}{5}}\div 243^{-\frac{1}{5}}

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

Evaluate:

\left(\dfrac{4}{9}\right)^{\frac{1}{3}} \times \left(\dfrac{4}{9}\right)^{\frac{1}{3}}

\left(\dfrac{3}{5}\right)^{\frac{3}{4}} \times \left(\dfrac{3}{5}\right)^{-\frac{7}{4}}

\left(\dfrac{8}{27}\right)^{\frac{4}{3}} \div \left(\dfrac{8}{27}\right)^{\frac{2}{3}}

\left(\dfrac{343}{216}\right)^{\frac{1}{3}} \div \left(\dfrac{343}{216}\right)^{-\frac{1}{3}}

Evaluate:

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

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

Evaluate:

\left(16^{\frac{1}{3}}\right)^{\frac{3}{4}}

\left(64^{\frac{1}{6}}\right)^{\frac{6}{3}}

\left(243^{\frac{2}{3}}\right)^{\frac{6}{5}}

\left(256^{\frac{1}{5}}\right)^{\frac{5}{8}}

\left(25^{\frac{1}{5}}\right)^{-\frac{5}{2}}

\left(81^{-\frac{1}{3}}\right)^{\frac{9}{4}}

\left(128^{\frac{6}{7}}\right)^{-\frac{3}{2}}

\left(27^{-\frac{5}{4}}\right)^{-\frac{4}{3}}

Evaluate:

\left(\dfrac{4^{4}}{81}\right)^{\frac{1}{4}}

\left(\dfrac{3^{6}}{64}\right)^{\frac{1}{6}}

\left(\dfrac{2^{3}}{27}\right)^{\frac{2}{3}}

\left(\dfrac{27^{2}}{64}\right)^{\frac{5}{6}}

\left(\dfrac{5^4}{256}\right)^{-\frac{1}{4}}

\left(\dfrac{11^5}{243}\right)^{-\frac{1}{5}}

\left(\dfrac{16}{5^4}\right)^{-\frac{3}{4}}

\left(\dfrac{2^8}{81}\right)^{-\frac{5}{4}}

Evaluate:

\left(\dfrac{8}{27}\right)^{\frac{1}{3}}

\left(\dfrac{81}{16}\right)^{\frac{5}{4}}

\left(\dfrac{64}{125}\right)^{\frac{2}{3}}

\left(\dfrac{625}{16}\right)^{\frac{1}{4}}

\left(\dfrac{64}{27}\right)^{\frac{2}{3}}

\left(\dfrac{125}{216}\right)^{\frac{2}{3}}

\left(\dfrac{27}{8}\right)^{\frac{4}{3}}

\left(\dfrac{1000}{27}\right)^{\frac{4}{3}}

Evaluate:

\left(\dfrac{16}{9}\right)^{-\frac{1}{2} }

\left(\dfrac{25}{64}\right)^{-\frac{1}{2} }

\left(\dfrac{27}{8}\right)^{-\frac{4}{3} }

\left(\dfrac{16}{81}\right)^{-\frac{3}{4} }

\left(\dfrac{64}{125}\right)^{-\frac{2}{3} }

\left(\dfrac{16}{625}\right)^{-\frac{3}{4} }

\left(\dfrac{64}{27}\right)^{-\frac{2}{3} }

\left(\dfrac{729}{343}\right)^{-\frac{2}{3} }

Write the following in surd form:

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

a^{\frac{3}{2} }

y^{ - \frac{1}{2} }

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

2x^{\frac{5}{4}}

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

y^{1.5}

p^{ - \frac{4}{3} }

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

\sqrt{x}

\sqrt[6]{x}

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

\left(\sqrt[3]{x}\right)^{2}

\sqrt[4]{x^3}

\left(\sqrt[7]{x}\right)^{6}

\sqrt[4]{x^8}

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

\sqrt{\sqrt{x}}

\sqrt[3]{x^{ 6 a}}

\dfrac{1}{\sqrt{x^{ - 3 }}}

x\sqrt{x}

\dfrac{1}{\sqrt[4]{x^{5}}}

\sqrt[5]{\sqrt{x}}

x^2\times \sqrt[3]{x}

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

\left(\sqrt{b}\right)^8

\sqrt{m^{6}}

\sqrt[4]{a^{5}}

\dfrac{1}{\sqrt[5]{a^{6}}}

\sqrt[3]{x^{6}}

\sqrt[3]{m^{3}}

\sqrt{ j^{2} x^{4}}

\left(\sqrt[4]{ x^{5} y^{3}}\right)^{24}

\left( 4 a^{8}\right)^{\frac{1}{2}}

\left( 625 u^{16} v^{12}\right)^{\frac{1}{2}}

8 b^{\frac{3}{4}} \div 2 b^{\frac{2}{3}}

y^{3} \times \sqrt[3]{y}

\sqrt{\left( 2 x + 9\right)^{2}}

\sqrt[3]{ 9^{3} x^{18} y^{12}}

\left(\dfrac{x^{15}}{1024}\right)^{\frac{2}{5}}

\sqrt{\dfrac{36 x^{18}}{y^{20}}}

\sqrt{ 16 y^{2} + 24 y + 9}

\sqrt{ x^{2} y^{2} + 18 x y^{2} + 81 y^{2}}

Outcomes

ACMNA264 (10a)

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

2.06 Fractional indices

Outcomes

ACMNA264 (10a)

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