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1.01 Index laws

Worksheet
Product and division laws
1

Simplify the following, giving your answer in index form:

a

2^{2} \times 2^{2}

b

4 y^{3} \times 6 y

c
3x^5 \times 8x^2
d

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

e

\dfrac{x^{6}}{4 x^{4}}

f

\dfrac{u^{ 2 x + 1}}{u^{x}}

g

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

h

\dfrac{4 n^{3} \times 4 n^{4}}{16}

i

p^{18} \div p^{8} \div p^{5}

j

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

k

m^{9} \div m^{5} \times m^{4}

l

\left( - \dfrac{7}{5} \right)^{m} \times \left( - \dfrac{7}{5} \right)^{n}

2

Rewrite the following expressions without brackets:

a

\left(\dfrac{a}{b}\right)^{3}

b
(xy)^7
c
\left( 2x \right)^4
d
\left( \dfrac{xy}{5z}\right)^2
3

Complete the statement below:

15 j^{14} \div \left(⬚\right) = 5 j^{7}

Negative indicies and the zero index
4

Simplify the following, giving your answer in positive index form:

a

m^{2} \times m^{ - 7 }

b

\left( 4 m^{ - 10 }\right)^{4}

c

\left( 4 m^{ - 8 }\right)^{ - 3 }

d

\dfrac{12 x^{3}}{4 x^{7}}

e

\dfrac{9 x^{3}}{3 x^{ - 4 }}

f

\left(\dfrac{a^{3}}{b^{3}}\right)^{ - 5 }

g

\left(\dfrac{2 h}{3}\right)^{ - 4 }

h

\dfrac{\left(m^{ - 3 }\right)^{ - 1 } \times \left(m^{4}\right)^{ - 3 }}{m^{3} \times m^{4}}

5

Express the following in simplest form without negative indices:

a

\left(\dfrac{a}{b}\right)^{ - 5 }

b

2 p^{4} q^{ - 2 } \times 5 p^{ - 4 } q^{ - 5 }

c

\dfrac{5 p^{5} q^{ - 4 }}{40 p^{5} q^{6}}

d
10x^3y^2z \div 2x^5y^9
e
2^{15} \div 2^{ - 5 }
6

Express the fraction \dfrac{m}{n^{4}} as a product using negative indices.

7

Simplify:

a

a^{0}

b
8x^0
c

\left( 2 \times 13\right)^{0}

d

\left(a^{0}\right)^{79}

e

\left( 13 x^{7}\right)^{0} + 13^{0} - 13 h^{0}

f

\left( 7 m^{0} + 4\right)^{2}

Power of a power
8

Simplify the following:

a

\left(w^{3}\right)^{4}

b

\left( 3 y^{6}\right)^{2}

c
\left(\dfrac{x^4}{2y^5}\right)^3
d

\left(u^{x + 1}\right)^{3}

e

\left( 2 y^{4}\right)^{2} \times \left( 2 y^{2}\right)^{3}

f

\left(\dfrac{1}{b}\right)^{3}

g

\left( - \dfrac{5 a}{2} \right)^{3}

h

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

i

\dfrac{\left(x^{3}\right)^{2}}{x^{3}}

j

\left( 2^{3} \div 3^{2}\right)^{3}

k

\dfrac{3^{ 4 a + 2} \times 3^{1 + 6 a}}{\left(3^{3}\right)^{ 3 a - 1}}

l

\dfrac{81^{ 7 a - 4} \times 9^{ 3 a + 2}}{27^{3 - 3 a}}

9

Find the value of a and b in the following equation:

\dfrac{v^{18}}{w^{24}} = \left(\dfrac{v^a}{w^{4}}\right)^b

10

Write \left(16^{p}\right)^{4} in the form a^b, where a is a prime number.

Fractional indices
11

Write the following in surd form:

a
x^{\frac{1}{3}}
b
y^{ - \frac{1}{2} }
c
x^{\frac{3}{4}}
d
4a^{-\frac{2}{5}}
12

Write the following in index form:

a

\sqrt{x}

b

\sqrt[6]{x}

c

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

d

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

e

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

f
\sqrt{m^{6}}
g
\sqrt[4]{a^{5}}
h
\dfrac{1}{\sqrt[5]{a^{6}}}
13

Simplify:

a
\sqrt{\left(\dfrac{x^{4}}{4}\right)}
b
\sqrt{\left( 5^{2} x^{14} y^{20}\right)}
14

Patricia's working out for evaluating 36^{\frac{3}{2}} is shown to the right:

a

There is an error in her working. In which line did Patricia make an error?

b

What should she have written in this line?

c

Hence, evaluate 36^{\frac{3}{2}}.

\begin{aligned}36^{\frac{3}{2}}&=\left(36^{\frac{1}{2}}\right)^3\\ &= \left(\dfrac{\sqrt{36}}{2}\right)^3\\ &= 3^3\\&= 27\end{aligned}

15

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

a

Complete the following working:

m^{\frac{1}{4}} = (m^⬚)^{\frac{1}{2}}

\text{ }= \sqrt{⬚}^{\frac{1}{2}}

b

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

16

Solve the following equation for k:

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

17

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

a

Express it in simplest index form.

b

Express it in surd form.

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MA11-1

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