Fractional and Negative Indices worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

10 questions 40 marks Sheet 1

2 marks
(a)
\(m^{ 2 } \times m^n = m^{ 8 }\)

Calculate the value of \(n\).

(b)
\(m^{ 8 } \div m^n = m^{ 4 }\)

Calculate the value of \(n\).
4 marks
(a)
\(n^{\frac{1}{2}} = 6\)

Find the value of \(n\).

(b)
\(m^{\frac{1}{3}} = 4\)

Find the value of \(m\).
5 marks
(a)
Work out \(243^{\frac{1}{ 5 } }\)

(b)
Work out \(16^{\frac{ 3 }{ 4 } }\)

(c)
Work out \(49^{\frac{3}{2} }\)
5 marks
(a)
Write \(\sqrt{ t }\) in the form \(t^n\)

(b)
Write \(\frac{ 7 }{ t^{ 5 } }\) in the form \(at^n\)

(c)
Write \(t\sqrt{ t }\) in the form \(t^n\)
4 marks
(a)
Work out \(64^{-\frac{1}{2} }\)

Give your answer as a fraction.

(b)
Work out \(16^{-\frac{ 3 }{ 4 } }\)

Give your answer as a fraction.
4 marks
(a)
Work out the value of \(\left(\frac{ 4 }{ 5 }\right)^{-3 }\)

(b)
Work out the value of \(\left(\frac{ 9 }{ 49 }\right)^{-\frac{1}{2} }\)
4 marks
(a)
Solve \(4^x = \frac{1}{ 256 }\)

(b)
Solve \(8^x = 2\)
6 marks
(a)
Simplify fully \(\frac{ (x^{ 2 })^{ 2 } }{ x^{ 15 } }\)

Give your answer as a single power of \(x\).

(b)
Simplify fully \(\left(64x^{ 9 }\right)^{\frac{1}{ 3 } }\)

(c)
Simplify fully \(x^{\frac{ 7 }{2} } \times x^{\frac{ 7 }{2} }\)
2 marks
(a)
\(t^{ 3 } \times t^n = t^{ 11 }\)

Find the value of \(n\).

(b)
\(t^{ 9 } \div t^n = t^{ 3 }\)

Find the value of \(n\).
4 marks
(a)
\(n^{\frac{1}{2}} = 5\)

Find the value of \(n\).

(b)
\(m^{\frac{1}{3}} = 4\)

Find the value of \(m\).

Answers

(a) \(6\)
(b) \(4\)
(a) \(36\)
(b) \(64\)
(a) \(3\)
(b) \(8\)
(c) \(343\)
(a) \(t^{\frac{1}{2} }\)
(b) \(7t^{-5 }\)
(c) \(t^{\frac{3}{2} }\)
(a) \(\frac{1}{ 8 }\)
(b) \(\frac{1}{ 8 }\)
(a) \(\frac{ 125 }{ 64 }\)
(b) \(\frac{ 7 }{ 3 }\)
(a) \(x = -4\)
(b) \(x = \frac{1}{ 3 }\)
(a) \(x^{-11 }\)
(b) \(4x^{ 3 }\)
(c) \(x^{ 7 }\)
(a) \(8\)
(b) \(6\)
(a) \(25\)
(b) \(64\)