5. Indices and Surds
iGCSE (2021 Edition)
5.02 Index laws
Worksheet
1
Write the following expressions in simplest index form:
a
2^{12} \times 2^{9}
b
2^{8} \times 11^{8}
c
11^{12} \div 11^{8}
d
21^{5} \div 3^{5}
e
\left(5^{12}\right)^{4}
f
15^{17} \div 15^{8} \div 15^{5}
g
\left(23^{8}\right)^{9} \times 23^{7}
h
\dfrac{\left(17^{5}\right)^{8}}{17^{32}}
i
\dfrac{19^{9} \times 19^{4}}{19^{8}}
j
\dfrac{12^{6}}{12^{4}} \times 12^{5}
k
\dfrac{\left(13^{5}\right)^{2} \times 13^{3}}{13^{5}}
l
\dfrac{\left(15^{9}\right)^{5} \times 15^{7}}{15^{25}}
2
Complete the following statements:
a
11^{11} \times 11^{⬚} = 11^{19}
b
5^{11} \times \left(⬚\right)^{11} = 35^{11}
c
97^{⬚} \div 97^{22} = 97^{12}
d
55^{6} \div \left(⬚\right)^{6} = 11^{6}
e
\left(11^{4}\right)^{⬚} = 11^{12}
f
7^{8} \times 7^{⬚} = 7^{14}
g
2^{10} \times \left(⬚\right)^{10} = 10^{10}
h
19^{⬚} \div 19^{18} = 19^{20}
i
60^{4} \div \left(⬚\right)^{4} = 12^{4}
j
\left(13^{8}\right)^{⬚} = 13^{16}
3
Evaluate the following expressions:
a
6^{5} \times 6^{3}
b
7^{3} \times 3^{3}
c
4^{8} \div 4^{3}
d
\left(5^{4}\right)^{2}
e
35^{5} \div 5^{5}
f
2^{4} \times 4^{4}
g
11^{18} \div 11^{9} \div 11^{7}
h
\left(3^{3}\right)^{2}
i
\dfrac{6^{5} \times 6^{9}}{6^{12}}
j
7^{27} \div 7^{30} \div 7^{3}
k
\dfrac{12^{10} \times 12^{4}}{12^{11}}
l
\dfrac{\left(6^{8}\right)^{6}}{6^{46}}
4
Write the following expressions in simplest index form:
a
\left( - 11 \right)^{10} \times \left( - 11 \right)^{3}
b
\left( - 7 \right)^{8} \times 3^{8}
c
\left(-5 \right)^{2} \times 3^{2}
d
\left( - 3 \right)^{12} \div \left( - 3 \right)^{5}
e
\left( - 12 \right)^{20} \div \left( - 12 \right)^{19}
f
\left( - 30 \right)^{50} \div \left( - 30 \right)^{47}
g
\left( - 48 \right)^{3} \div \left(-6 \right)^{3}
h
\left( - 33 \right)^{11} \div \left( - 3 \right)^{11}
i
\left( - 35 \right)^{5} \div 5^{5}
j
\left( - 42 \right)^{2} \div 7^{2}
5
Complete the following statements:
a
11^{3} \times \left(⬚\right)^{3} = \left( - 77 \right)^{3}
b
\left( - 5 \right)^{11} \times \left(⬚\right)^{11} = 15^{11}
c
\left( - 5 \right)^{⬚} \div \left( - 5 \right)^{31} = \left( - 5 \right)^{19}
d
\left( - 14 \right)^{13} \div \left(⬚\right)^{13} = \left( - 7 \right)^{13}
e
\left( - 3 \right)^{⬚} \div \left( - 3 \right)^{39} = \left( - 3 \right)^{10}
f
\left( - 5 \right)^{4} \times \left(⬚\right)^{4} = \left( - 60 \right)^{4}
g
\left( - 3 \right)^{7} \times \left(⬚\right)^{7} = \left( 6 \right)^{7}
h
\left( {⬚} \right)^3 \div \left( - 3 \right)^{3} = \left( 7 \right)^{3}
i
\left(-33\right)^{3} \div \left( ⬚ \right)^{3} = -11^{3}
j
\left(⬚\right)^{7} \div \left( - 5 \right)^{7} = 11^{7}
6
Evaluate the following expressions:
a
\left( - 4 \right)^{11} \div \left( - 4 \right)^{7}
b
\left( - 2 \right)^{3} \times \left( - 2 \right)^{3}
c
\left( - 3 \right)^{3} \times \left( - 3 \right)^{2}
d
4^{3} \times \left( - 5 \right)^{3}
e
\left( - 3 \right)^{8} \div \left( - 3 \right)^{5}
f
15^{5} \div \left( - 3 \right)^{5}
g
2^3\times \left(-3\right)^3
h
\left(-14\right)^{11}\div 2^{11}
i
\left (-7\right)^{2} \times 5^{2}
j
\left(-9\right)^{4} \times \left(-3\right)^{4}
k
\left(-100\right)^{6} \div 50^{6}
l
60^{3} \div \left(-3\right)^{3}
7
Write the following in simplest index form:
a
\left(\dfrac{1}{3}\right)^{4}
b
\left(\dfrac{3}{8}\right)^{3}
c
\left(\dfrac{4}{16}\right)^{8}
d
\left(\dfrac{15}{6}\right)^{2}
e
\left(\dfrac{10}{33}\right)^{5}
f
\left(\dfrac{2}{35}\right)^{6}
g
\left(\dfrac{5}{18}\right)^{3}
h
\left(\dfrac{29}{41}\right)^{7}
i
\left(\dfrac{11}{13}\right)^{9}
j
\left(\dfrac{20}{3}\right)^{2}
k
\left(\dfrac{17}{4}\right)^4
l
\left(\dfrac{31}{50}\right)^5
8
Complete the following statements:
a
\dfrac{1}{27} = \left(\dfrac{1}{3}\right)^{⬚}
b
\dfrac{64}{27} = \left(\dfrac{4}{3}\right)^{⬚}
c
\dfrac{27}{8} = \left(\dfrac{3}{⬚}\right)^{3}
d
\dfrac{⬚}{16} = \left(\dfrac{1}{4}\right)^{2}
e
\dfrac{27}{8} = \left(\dfrac{3}{2}\right)^{⬚}
f
\dfrac{⬚}{625}=\left(\dfrac{2}{5}\right)^{4}
g
\dfrac{81}{100} = \left(\dfrac{⬚}{10}\right)^{2}
h
\dfrac{256}{⬚} = \left(\dfrac{4}{5}\right)^{4}
i
\dfrac{25}{9} = \left(\dfrac{5}{3}\right)^{⬚}
j
\dfrac{⬚}{144} = \left(\dfrac{11}{12}\right)^{2}
k
\dfrac{125}{27} = \left(\dfrac{⬚}{3}\right)^3
l
\dfrac{32}{729} = \left(\dfrac{2}{3}\right)^{⬚}
9
Evaluate the following expressions in fully simplified fraction:
a
\left(\dfrac{1}{3}\right)^{2}
b
\left(\dfrac{3}{5}\right)^{3}
c
\left(\dfrac{1}{2}\right)^{5}
d
\left(\dfrac{7}{11}\right)^{2}
e
\left(\dfrac{3}{8}\right)^{3}
f
\left(\dfrac{12}{13}\right)^2
g
\left(\dfrac{5}{9}\right)^{3}
h
\left(\dfrac{2}{3}\right)^{4}
i
\left (\dfrac{4}{5}\right)^{6}
j
\left(\dfrac{2}{5}\right)^{3}
k
\left(\dfrac{5}{7}\right)^{2}
l
\left(\dfrac{10}{14}\right)^3
10
State whether the following fractions are equal to \left(\dfrac{1}{2}\right)^{3} :
a
\dfrac{3}{6}
b
\dfrac{1}{2^{3}}
c
\dfrac{3}{2^{3}}
d
\dfrac{1}{8}
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