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AustraliaVIC
Level 10

1.03 Negative indices

Worksheet
Negative indices
1

Is 2^{ - 3 } less than or greater than 1?

2

Express the following with positive indices:

a

a^{ - 9 }

b

\dfrac{1}{a^{ - n }}

c

\dfrac{a^{ - 9 }}{4}

d

\dfrac{a^{ - n }}{b^{ - m }}

e

p^{ - 2 }

f

3 x^{ - 4 }

g

7 x^{ - 9 }

h

p^{-2}q^3

i

8p^{-3}

j

2x^{-8}y^3

k
\dfrac{b^{-7}}{6c^{-3}}
l
m^{-5}n^{-4}p^4
m
3a^{4}b^{-5}c^{-7}
n
4^{-2}k^{-3}l^{7}
o
\dfrac{f^{-7}}{g^{-2}h^8}
p
\dfrac{s^4 t^{-9}}{5u^{-6}}
3

Express the following without fractions:

a
\dfrac{1}{u^{4}}
b
\dfrac{2}{r^6}
c
\dfrac{x}{y^{2}}
d
\dfrac{4}{p^6q^7}
4

Simplify the following, giving your answers with positive indices:

a

5 y^{9} \times 4 y^{ - 3 }

b

7 a^{4} \times 4 a^{-2}

c

5x^4\times \left(-3x^{-8}\right)

d

3y^{-2}\times 4y^{-3}

e

2h^{-4}\times 4h^{11}

f

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

g

-4y^2\times \left(-4y^{-5}\right)

h

\left(5mp\right)^2\times mp^{-2}

i

\dfrac{9 x^{2}}{3 x^{9}}

j

\dfrac{15x^3}{5x^7}

k
12x^5 \div 4x^{-3}
l
9c^6 \div 6c^8
m
\dfrac{15h^2}{12h^{-7}}
n
\dfrac{12x^5}{4x^{-3}}
o
\dfrac{21p^{-4}}{14p^{-3}}
p
\dfrac{36w^{-6}}{16w^4}
5

Simplify the following, giving your answers with positive indices:

a

\left( 2 m\right)^{ - 3 }

b

\left(4m^{-6}\right)^4

c

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

d

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

e

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

f

\left(\dfrac{x^{5}}{y^{4}}\right)^{ - 1 }

g

\left(\dfrac{x^{7}}{y^{9}}\right)^{ - 4 }

h

\dfrac{20 x^{3}}{4 x^{ - 2 }}

i

\dfrac{10 x^{ - 7 }}{2 x^{ - 3 }}

j

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

k

\left(\dfrac{p^{3}}{q^{7}}\right)^{ - 1 }

l

\left(\dfrac{x^{-4}}{y^{-8}}\right)^{ - 2 }

6

Solve the following equations for n:

a

\dfrac{1}{25} = 5^{n}

b

\dfrac{1}{8} = 2^{n}

c

\left( x^{3} y^{ - 5 }\right)^{n} = x^{ - 12 } y^{20}

d

\left( a^{-5} b^{ 3 }\right)^{n} = a^{ 15 } b^{-9}

7

Complete the following:

\left(x^4 y^⬚\right)^⬚=\dfrac{y^{12}}{x^{16}}
8

A student was writing 5a^{-1} without negative indices and wrote 5a^{-1}=\dfrac{1}{5a}. Explain why their working is incorrect, and write the correct answer.

9

Simplify:

a

2 y^{6} \times 4 y^{7} \times 4 y^{ - 5 }

b

6 y^{7} \times 2 y^{ - 5 } \times 5 y^{3}

10

Simplify the following, giving your answers with positive indices:

a
\dfrac{2x^2 y^{-4}}{7z^{-3}}\times\dfrac{5x^{-5}z^6}{3y^3}
b
\left(\dfrac{3x^{-2}}{10y^4 z^{-1}}\right)^{-2}\div \left(\dfrac{8x^{-5} z^2}{15y^2}\right)^{-1}
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VCMNA330

Simplify algebraic products and quotients using index laws.

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