5. Indices and Surds
iGCSE (2021 Edition)
5.05 Variables and indices
Worksheet
1
Simplify:
a
y^{3} \times y^{2}
b
x^{6} \times 8 x^{3}
c
3 y^{6} \times 4 y
d
x^{y} x^{z}
e
a^{7} \times -6 a^{5}
f
-2 a^{4} \times 5 a
g
-b^{7} \times -b^{6}
h
a^{b} a^{c}
i
5y^{5} \times 6y^{3}
j
-8 u^{3} \times -3u
k
-4b^{6} \times -5b^{7}
l
x^{a}x^{b} x^{c}
2
Complete the following statements:
a
b^{4} \times b^{⬚} = b^{7}
b
7 x^{15} \times ⬚ = 35 x^{27}
c
c^{10} \times c^{⬚} = c^{8}
d
4 x^{20} \times ⬚ = 24 x^{25}
3
Simplify:
a
a^9\div a^5
b
\dfrac{x^{11}}{5 x^{8}}
c
\dfrac{6 m^{15}}{m^{4}}
d
b^6\div b^2
e
\dfrac{j^{9}}{j^{2}}
f
x^{7} \div x^{4}
g
\dfrac{f^{8}}{f^{5}}
h
s^{13}\div s^4
i
\dfrac{4 k^{11}}{2k^{6}}
j
\dfrac{8g^{12}}{g^{5}}
k
6p^{13} \div 2p^{10}
l
15l^8\div 3l^2
4
Complete the following statements:
a
b^{9} \div b^{⬚} = b^{5}
b
x^{⬚}\div x^3=x^2
c
\dfrac{g^⬚}{g^4}=g^7
d
\dfrac{k^7}{k^⬚}=k^3
5
Consider the expression \left(p^{3}\right)^{2}.
a
State whether the following expressions are equivalent to \left(p^{3}\right)^{2}:
i
\left( p \times p \times p\right) \times \left( p \times p \times p\right)
ii
p^{3} \times p^{3}
iii
\left( p \times p \times p\right)^{2}
iv
p^{3} \times p^{2}
v
\left( p \times p \times p\right) \times \left( p \times p\right)
b
State whether the following equations are true or false:
i
\left(p^{3}\right)^{2} = p^{3 + 2}
ii
\left(p^{3}\right)^{2} = p^{3 \times 2}
c
Complete the following:
\left(p^{3}\right)^{2} = p^{⬚}6
Simplify:
a
\left(j^{3}\right)^{6}
b
\left(w^{2}\right)^{4}
c
\left(t^4\right)^3
d
\left(3a^4\right)^3
e
\left(5h^7\right)^3
f
\left(xy^2\right)^5
g
\left(\dfrac{2}{h^2}\right)^4
h
\left(\dfrac{a^2}{b^5}\right)^3
i
\left(\dfrac{3x^2}{y^5}\right)^3
j
\left( - x^{9} \right)^{4}
k
\left(-5x^4\right)^3
l
\left(\dfrac{-2a^4}{3b^2}\right)^3
7
Find the value of a and b in the following equation:
\dfrac{v^{18}}{w^{24}} = \left(\dfrac{v^a}{w^{4}}\right)^b
8
Simplify:
a
18 a^{0}
b
\left(f^{0}\right)^{9}
c
\left(g^{12}\right)^{0}
d
\left( 6 a\right)^{0}
e
8r^0+\left(2q^3\right)^0
f
\left(\dfrac{2}{x^2}\right)^0
g
\dfrac{3r^0}{t^2}
h
8k^0\div\left(4x^2\right)^0
9
Complete the following statements:
a
b^{11} \div b^{11} = b^{⬚}
b
\dfrac{h^{⬚}}{h^{8}} = h^{0}
c
\dfrac{v^{9}}{v^{⬚}} = v^{0}
d
y^⬚\times y^5=y^5
10
Simplify:
a
\left( x^{6} y^{3}\right)^{4}
b
\dfrac{y^{7} \times y^{6}}{y^{3} \times y^{2}}
c
p^{5} \times \left(p^{4}\right)^{3}
d
\left( 2 y^{2}\right)^{3}
e
m^{5} \div m^{2} \times m^{5}
f
p^{9} \div p^{5} \div p^{2}
g
\left( - 2 x^{3} \right)^{4}
h
\dfrac{6 p^{5} \times 4 p^{7}}{8 p^{3}}
i
\left( 7 x^{2}\right)^{0} - \left( 8 x^{6}\right)^{0}
j
\left( 12 x^{4}\right)^{0} + 12^{0} - 12 h^{0}
k
\left( 4 h^{0}\right)^{2} + 18 \div \left( 3 g^{0}\right)
l
\left(\left(x^{2}\right)^{6}\right)^{5}
m
\left( 4 u^{5} v^{2}\right)^{2}
n
\left( 3 a^{2} b^{5} c\right)^{4}
o
\left( 3 y^{5}\right)^{2} \times \left( 5 y^{2}\right)^{3}
p
\dfrac{\left(x^{4}\right)^{2}}{x^{5}}
q
\dfrac{\left( 2 x^{2} y^{0}\right)^{4}}{x^{5}}
r
\left(c^{10}\right)^{11} \div \left(c^{8}\right)^{3}
s
\dfrac{24 a^{3}}{\left( 2 a\right) \left( 4 b\right)}
t
\left( 10 x^{2} \times 10x^{5}\right)^{0} - 10 x^{0}
u
\dfrac{3^{ 4 a + 2} \times 3^{1 + 6 a}}{\left(3^{3}\right)^{ 3 a - 1}}
v
\dfrac{81^{ 7 a - 4} \times 9^{ 3 a + 2}}{27^{3 - 3 a}}
11
Write \left(16^{p}\right)^{4} in the form a^b, where a is a prime number.
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