5. Indices and Surds
iGCSE (2021 Edition)
5.03 Negative indices
Worksheet
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} |
| 32 | 16 |
b
| \quad10^{5} | \quad10^{4} | \enspace10^{3} | 10^{2} | 10^{1} | 10^{0} | 10^{-1} |
| 100\,000 | 10\,000 |
c
| 3^{3} | 3^{2} | 3^{1} | 3^{0} | 3^{-1} | 3^{-2} | 3^{-3} |
| 27 | 9 |
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}}
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}
10
Evaluate the following expressions:
a
\dfrac{\left(14^{3}\right)^{4}}{14^{4} \times 14^{8}}
b
\dfrac{3^{ - 9 } \times 3^{ - 7 }}{3^{ - 14 }}
c
\dfrac{5^{7}\times 5^{-8}}{5^{-4}}
d
\dfrac{\left(-7\right)^{ - 11 } \times \left(-7\right)^{ - 4 }}{\left(-7\right)^{ - 14 }}
11
Simplify the following expressions:
a
\dfrac{\left(5^{2}\right)^{9} \times 5^{6}}{5^{40}}
b
\dfrac{\left(19^{2}\right)^{3}}{19^{ - 3 } \times 19^{ - 9 }}
c
\dfrac{-\left(8^{5}\right)^{2}}{8^{ - 2 } \times 8^{ - 12 }}
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