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iGCSE (2021 Edition)

1.03 Negative indices

Worksheet
Negative indices
1

Consider the following expressions:

i

Identify the base.

ii

Identify the power.

a
10^{ - 7 }
b
2^{ - 4 }
c
13^{-10}
d
\left(-5\right)^{-8}
2

Complete the following tables:

a
2^{5}2^{4}2^{3}2^{2}2^{1}2^{0}2^{-1}
3216
b
\quad10^{5}\quad10^{4}\enspace10^{3}10^{2}10^{1}10^{0}10^{-1}
100\,00010\,000
c
3^{3}3^{2}3^{1}3^{0}3^{-1}3^{-2}3^{-3}
279
3

Express the following expressions with a positive index:

a
6^{ - 10 }
b
73^{ - 14 }
c
\left( - 9 \right)^{ - 7 }
d
9^{ - 1 }
e
17^{ - 6 }
f
55^{ - 1 }
g
\left( - 12 \right)^{ - 8 }
h
-45^{ - 5 }
i
-8^{ - 11 }
j
\left(-20\right)^{-3}
k
7^{-6}
l
\left(-5\right)^{-1}
4

Express the following expressions with a negative index:

a
\dfrac{1}{3}
b
\dfrac{1}{37}
c
\dfrac{1}{5}
d
\dfrac{1}{4^{7}}
e
\dfrac{1}{-15^{3}}
f
\dfrac{1}{10^{5}}
g
\dfrac{1}{\left(-24\right)^{10}}
h
\dfrac{1}{25^{3}}
i
\dfrac{1}{13^{11}}
j
\dfrac{1}{7^{8}}
k
\dfrac{1}{16^{12}}
l
\dfrac{1}{\left(-45\right)^{7}}
5

Complete the following statements:

a
\dfrac{1}{3^{2}} = 3^{⬚}
b
\dfrac{1}{6^{5}} = 6^{⬚}
c
\dfrac{1}{27} = 3^{⬚}
d
\dfrac{1}{5^{3}} = 5^{⬚}
e
\dfrac{1}{7^{11}} = 7^{⬚}
f
\dfrac{1}{64} = 4^{⬚}
g
\dfrac{1}{-9^{4}} = -9^{⬚}
h
\dfrac{1}{-32} = -2^{⬚}
6

Simplify the following expressions:

a
5^{11} \div 5^{ - 3 }
b
7^{ - 7 } \div 7^{5}
c
7^{ - 3 } \times 7^{ - 4 }
d
5^{ - 4 } \div 5^{ - 9 }
e
9^{0} \times 9^{ - 12 }
f
5^{13}\div 5^{-9}
g
10^{2} \div 10^{3}
h
100^{25} \div 100^{26}
i
\dfrac{2^{3}}{2^{5}}
j
\dfrac{\left(5^{2}\right)^{9} \times 5^{6}}{5^{40}}
k
\dfrac{\left(19^{2}\right)^{3}}{19^{ - 3 } \times 19^{ - 9 }}
l
\dfrac{-\left(8^{5}\right)^{2}}{8^{ - 2 } \times 8^{ - 12 }}
7

Evaluate:

a
4 \times 3^{ - 2 } + 8^{0}
b
6 \div 2^{ - 1 } + 4
c
\left(2+3^{-2}\right)\times 2^{-1}
d
\left(\dfrac{2}{3}\right)^{-3}\div 6^{-1}
8

Answer the following questions:

a

What is 0^4 equal to?

b

Explain why 0^{-4} is undefined.

9

Evaluate the following expressions:

a
6^{7} \times 6^{ - 7 }
b
4^{8} \times 4^{ - 6 }
c
3^{ - 8 } \times 3^{11}
d
5^{5} \times 5^{ - 7 }
e
4^{-4} \times 2^{-4}
f
12^{13} \div 12^{7} \div 12^{8}
g
5^{-2} \times 3^{-2}
h
\left(-4\right)^{3} \times \left(-4\right)^{-7}
i
7^{12} \div 7^{-7} \div 7^{17}
j
\dfrac{3^{ - 9 } \times 3^{ - 7 }}{3^{ - 14 }}
k
\dfrac{5^{7}\times 5^{-8}}{5^{-4}}
l
\dfrac{\left(-7\right)^{ - 11 } \times \left(-7\right)^{ - 4 }}{\left(-7\right)^{ - 14 }}
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