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Middle Years

1.05 Index laws

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Worksheet
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Worksheet
Product and division laws
1

Write the following using index notation: 3 \times u \times u \times u \times 5 \times v \times v \times v

2

Simplify the following:

a

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

b
2^{3} \times 2^{4}
c

4 y^{3} \times 6 y

d
c^{ 2 z} c^{z + 1}
e
3x^5 \times 8x^2
f

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

g
\dfrac{3^{6}}{3^{3}}
h

\dfrac{x^{6}}{3 x^{4}}

i

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

j
\dfrac{10 p^{6} \times 3 p^{10}}{15 p^{2}}
k

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

l
\dfrac{5^{ 2 x}}{5^{x + 1}}
m

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

n

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

o
\dfrac{4 j^{5} k^{9}}{4 j^{4} k^{6}}
p
\dfrac{12 a^{3x-4} b^{7}}{4 a^{2x+1} b^{6}}
3

Rewrite the following expressions without brackets:

a

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

b
(xy)^7
c
\left( 2x \right)^4
d
\left( -3x \right)^3
e

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

f

\left(\dfrac{-1}{2a}\right)^{2}

g

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

h
\left( \dfrac{xy}{5z}\right)^2
i
2 x^{2} \left( 3 x^{2} + 7 x^{5}\right)
j
10 u^{2} \left( 7 u^{4} + 8 u^{3}\right)
k
6 u^{7} \left( 9 u^{7} + 9 u^{6}\right)
l
3x^{4} \left( 5 - \dfrac{4}{x^2}\right)
4

Write the following as a single power of 3:

a
81\times 3^2
b
3^x\times 3^2
c
3^{x+1}\times 3^{5x}
d
\dfrac{27}{3^{2x}}
5

Complete the statement below:

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

Power of a power
6

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(\left(x^{3}\right)^{4}\right)^{5}
f

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

g

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

h
\dfrac{\left(k^{12}\right)^{2}}{\left(k^{4}\right)^{7}}
i

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

j
\left( 6 b^{4}\right)^{5} \div \left( 3 b^{1}\right)^{2}
k
\dfrac{\left(u^{x + 3}\right)^{3}}{u^{x + 1}}
l
\left(8^{ 3 u}\right)^{\left( 4 p + 5 q\right)}
m
\dfrac{\left( n^{8} r^{5}\right)^{5}}{\left( n^{4} r\right)^{5}}
n
\left(\dfrac{- 2 x^{2} y^{6}}{z^{2}}\right)^{3}
o
\left(x^{2}\right)^{4} \div \left(\dfrac{1}{x^{3}}\right)^{3}
p
\left(t^{6}\right)^{5} \div \left(\dfrac{t^{9}}{t^{7}}\right)^{3}
q

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

r

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

7

Write the following as a single power of 2:

a
8^x
b
4\times 16^x
c
\left(2^{x+1}\right)^2\times 2^{3x}
d
\dfrac{32}{8^{x-1}}
8

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

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

9

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

10

Use index laws to simplify the ratio \left(0.3\right)^{a}:\left(0.3\right)^{a + 2} to a ratio of whole numbers.

11

Express 165^{m} as the product of prime bases. Leave your answer in expanded index form.

12

Find the next term in the sequence\left( 2 x^{2}\right)^{2}, \left( 2 x^{2}\right)^{3},\left( 2 x^{2}\right)^{4}, \left( 2 x^{2}\right)^{5} \ldots

Leave your answer in the form a^{m} b^{n}.

13

A computer is downloading data at a rate of 3 t^{3} bytes per second. If it downloads 6 \left(t^{3}\right)^{5} bytes, find an expression for the number of seconds the download took. Leave your answer in simplest index form.

Negative indicies and the zero index
14

Write the following as a rational number in simplest form:

a
3^{ - 1 }
b
3^{ - 2 } \div 3^{3}
c
\left(\dfrac{1}{4}\right)^{ - 3 }
d
\left(\dfrac{5}{3}\right)^{ - 2 }
15

Write the following as a single power of 2:

a
8^{-1}
b
\dfrac{1}{4}
c
2\left(\dfrac{1}{2^x}\right)^{ - 3 }
d
\dfrac{4^m}{2^{-m}}
16

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

a

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

b
3 x^{ - 2 }
c

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

d

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

e

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

f

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

g
\left(5^{2}\right)^{ - p }
h
m^{ - 5 } n^{ - 4 } p^{4}
i
2 p^{5} q^{ - 7 } \times 6 p^{ - 9 } q^{9}
j
10x^3y^2z \div 2x^5y^9
k
2^{15} \div 2^{ - 5 }
l
\dfrac{9 x^{3}}{3 x^{ - 4 }}
m
\left(\dfrac{x^{2}}{y^{4}}\right)^{ - 1 }
n
\left(\dfrac{a^{3}}{b^{3}}\right)^{ - 5 }
o
\left( \dfrac{4}{5} u^{ - 3 }\right)^{4}
p
\dfrac{5 p^{5} q^{ - 4 }}{40 p^{5} q^{6}}
q
\dfrac{b^{3} \div b^{ - 7 }}{\left(b^{ - 4 }\right)^{ - 4 }}
r
\left(\dfrac{m^{7}}{m^{ - 10 }}\right)^{2} \times \left(\dfrac{m^{5}}{m^{2}}\right)^{ - 3 }
s
\dfrac{\left(m^{ - 3 }\right)^{ - 1 } \times \left(m^{4}\right)^{ - 3 }}{m^{3} \times m^{4}}
t
\dfrac{a^{2} \times a^{ - 5 } \times b^{ - 2 }}{\left( a \times a^{ - 5 } \times b^{2}\right)^{2}}
17

Simplify the following, giving your answers in negative index form:

a
\dfrac{1}{6^{4}}
b
\dfrac{4}{n^{2}}
c
\dfrac{25 x^{5}}{5 x^{9}}
18

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

19

Express the reciprocal of 8^{9} in:

a

Positive index form

b

Negative index form

20

State whether the following expressions are positive, negative or zero:

a
\left( - 3 \right)^{ - 2 }
b
- 3^{6}
c
\left( -3 \right)^{ 0 }
d
\left( -3 \right)^{ 5 }
21

If p > 0 and q < 0, which of the following statements is correct?

A

p^{ - 2 } < 0

B

- 2 q^3 < 0

C

q^{ - 2 } < 0

D

- \left(2 p\right)^2 < 0

22

Complete the statement below:

32^{ - 6 } = \left( 2^{2} \times \left(⬚\right)^{2}\right)^{ - 3 }

23

Simplify the following:

a
741^{0}
b

a^{0}

c
8x^0
d
\left(a^{0}\right)^{79}
e
\left( 7 m^{0} + 4\right)^{2}
f
5 \times 4^{ - 2 } + 9^{0}
g

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

h

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

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