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CanadaON
Grade 9

3.10 Negative exponents

Worksheet
Negative exponents
1

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

2

Express the following with positive exponents:

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 exponents:

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 exponents:

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 exponents 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 exponents:

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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Outcomes

9.B2.2

Analyse, through the use of patterning, the relationships between the exponents of powers and the operations with powers, and use these relationships to simplify numeric and algebraic expressions.

9.C1.4

Simplify algebraic expressions by applying properties of operations of numbers, using various representations and tools, in different contexts.

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