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2.03 Division properties of exponents and the zero exponent

Worksheet
Division properties of exponents and the zero exponent
1

Simplify the following, giving your answers in exponential form.

a

10^{8} \div 10^{6}

b

j^{8} \div j^{4}

c

a^{5} \div a^{3}

d

9 q^{6} \div \left( 4 q^{2}\right)

2

Simplify the following, giving your answers with positive exponents.

a

\dfrac{18^{32}}{18^{23}}

b

\dfrac{a^{10}}{a^{6}}

c

\dfrac{u^{13}}{u^{2}}

d

\dfrac{t^{7} r^{8}}{t^{5} r^{4}}

e

\dfrac{5 r^{7}}{r^{4}}

f

\dfrac{3 j^{13}}{4 j^{8}}

g

\dfrac{- x^{11}}{3 x^{4}}

h

\dfrac{9 j^{9} k^{8}}{4 j^{7}}

i

\dfrac{3 g^{8}}{2 g^{5} h^{6}}

j

\dfrac{9 j^{5} k^{7}}{j^{3} k^{4}}

k

\dfrac{j^{8} k^{9}}{7 j^{7} k^{8}}

l

\dfrac{3 m^{9} n^{4}}{8 m^{8} n^{2}}

m

\dfrac{4 p^{3} q^{9}}{2 p^{2} q^{7}}

n

\dfrac{2 p^{9} q^{6}}{6 p^{3} q^{3}}

o

\dfrac{3 j^{6} k^{4}}{2 j^{4} k^{2}}

p

\dfrac{y^{6}}{\left( 4 y\right)^{2}}

q

\dfrac{6 r^{3} s^{6}}{2 r^{2} s^{4}}

r

\dfrac{- 9 x^{13}}{3 x^{4}}

s

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

t

\dfrac{54 x^{11}}{6 x^{6}}

3

For the following, write the integer value or the term that should go in the space.

a

a^{9} \div a^{⬚} = a^{7}

b

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

c

x^{23} y^{10} \div \left(⬚\right) = x^{15} y^{5}

d

⬚ \div \left( g^{4} h^{3}\right) = g^{16} h^{5}

e

63 x^{18} \div ⬚ = 9 x^{14}

f

x^{21} y^{8} \div ⬚ = x^{14} y^{3}

g

\dfrac{8 r^{3} s^{8}}{⬚} = 2 r s^{2}

4

Simplify the following, giving your answers in exponential form:

a

\dfrac{\left( n^{8} r^{5}\right)^{5}}{\left( n^{4} r\right)^{5}}

b

\dfrac{30 x^{13} y^{26} z^{21}}{3 x^{11} y^{15} z^{6}}

c

\dfrac{15 x^{16} y^{25} z^{13}}{- 3 x^{14} y^{11} z^{2}}

d

\left(m^{12}\right)^{9} \div \left(m^{4}\right)^{2}

e

\left( - 240 u^{32} \right) \div \left( - 8 u^{9}\right) \div \left( - 5 u^{12} \right)

f

\dfrac{y^{7} \cdot y^{5}}{y^{3} \cdot y^{2}}

g

\dfrac{10 p^{6} \cdot 3 p^{10}}{15 p^{2}}

h

\dfrac{\left( 2 x^{4} y^{0}\right)^{2}}{x^{6}}

i

\dfrac{6 \left( w^{2} v\right)^{5} \cdot 27 y^{17} v^{7}}{\left( 3 w^{5} v^{3}\right)^{2} \cdot 2 y^{6} v}

j

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

k

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

l

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

m

\dfrac{45 b^{6}}{\left( 3 b\right) \left( 5 a\right)}

n

\dfrac{210 p^{22}}{7 p^{8}} \div \left( 5 p^{7}\right)

5

Fill in the missing value to complete the pattern.

k^5 = 1 \cdot k \cdot k \cdot k \cdot k \cdot k
k^4 = 1 \cdot k \cdot k \cdot k \cdot k
k^3 = 1 \cdot k \cdot k \cdot k
k^2 = 1 \cdot k \cdot k
k^1 = 1 \cdot k
k^0 = ⬚
6

Fill in the missing exponent to make the following equations true.

a

p^{12} \div p^{12} = p^{⬚}

b

\dfrac{b^{⬚}}{b^{11}} = b^{0}

c

\dfrac{w^{7}}{w^{⬚}} = w^{0}

7

Evaluate:

a

741^{0}

b

\left( - 983 \right)^{0}

c

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

8

Simplify:

a

18 a^{0}

b

q^{0}

c

11 a^{0}

d

\left( 6 a\right)^{0}

e

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

f

9 \cdot \left( 15 x^{6}\right)^{0}

g

\left( 3 m^{4}\right)^{3} \cdot m q^{0}

h

\dfrac{45 x^{7} y^{7}}{9 x^{7} y^{2}}

Additional questions
9

When simplifying the expression \dfrac{a^5\cdot a^4}{a^2}, Chad's first line of work was a^3\cdot a^2.

Is Chad's work correct? Explain.

10

When simplifying the expression \dfrac{a^5+a^4}{a^2}, Clarice's first line of work was a^3+a^2.

Is Clarice's work correct? Explain.

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Outcomes

MA.8.AR.1.1

Apply the Laws of Exponents to generate equivalent algebraic expressions, limited to integer exponents and monomial bases.

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