Practise Indices and roots. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
(a) Write down the value of \(25^{\frac{1}{2}}\)[1]
(b) Write down the value of \(\left(\frac{2}{9}\right)^{0}\)[1]
(c) Write down the value of \(4^{-1}\)[1]
Show the answer and mark scheme
(a)Answer: \(5\)
B1 for 5
Worked solution: \(25^{\frac{1}{2}} = \sqrt{25} = 5\)
(b)Answer: \(1\)
B1 for 1
Worked solution: Any non-zero number to the power 0 is 1.
(c)Answer: \(\frac{1}{4}\)
B1 for \(\frac{1}{4}\) oe
Worked solution: \(4^{-1} = \frac{1}{4}\)
Question 2Medium3 marks
Mia works out the value of \(8^{-\frac{2}{3}}\). Here is her working. \(8^{-\frac{2}{3}} = -\left(8^{\frac{2}{3}}\right) = -\left(\sqrt[3]{8}\right)^2 = -4\)
(a) Explain Mia's mistake.[1]
(b) Work out the correct value of \(8^{-\frac{2}{3}}\).[2]
Show the answer and mark scheme
(a)Answer: A negative power means the reciprocal, not a negative number: \(8^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}}\).
C1 for explaining that the negative index means 'one over' (the reciprocal), not a negative answer
Worked solution: \(x^{-n} = \frac{1}{x^n}\), so the minus sign in the index gives a reciprocal. Mia made the answer negative instead.
(b)Answer: \(\frac{1}{4}\)
M1 for \(\frac{1}{8^{\frac{2}{3}}}\) or \(8^{\frac{2}{3}} = 4\) or \(\left(\frac{1}{2}\right)^2\)
A1 for \(\frac{1}{4}\) oe
Worked solution: \(8^{\frac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4\), so \(8^{-\frac{2}{3}} = \frac{1}{4}\).
Question 3Hard6 marks
(a) Work out the value of \(20.25^{-\frac{3}{2}}\) Give your answer as a fraction in its simplest form.[3]
(b) Work out the value of \(25^{-\frac{1}{2}} \times 16^{-\frac{1}{4}}\)[3]
Show the answer and mark scheme
(a)Answer: \(\frac{8}{729}\)
M1 for one correct step, e.g. writing the base as the fraction \(\frac{81}{4}\)
M1 for a second correct step, e.g. dealing with the negative power, \(\left(\frac{4}{81}\right)^{\frac{3}{2}}\)