Worksheets · Foundation and Higher

Integer powers and roots

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 1 mark
(a) Work out the cube root of 216. (1)
Question 2Non-calculator · 5 marks
A positive integer is $2^3\times3^2\times5$ .
(a) Find the smallest positive integer multiplier that makes it a square number. (2)
(b) Find the square root of the resulting square number. (3)
Question 3Calculator · 5 marks
(a) Find the smallest positive integer n for which n/18 is a square integer and n/30 is a cube integer. Show how prime factors determine your answer. (5)

Answers and marks

(a) $6$

(a) $10$

(b) $60$

(a) $101250$

P1 Establishing $18=2\times 3^{2}$ or an equivalent valid method.
P1 Establishing $30=2\times 3\times 5$ or an equivalent valid method.
P1 For the exponent b of 3, b $-$ 2 must be even and b $-$ 1 a multiple of 3. The smallest possible b is 4. For the exponent c of 5, c must be even and c $-$ 1 a multiple of 3, so the smallest c is 4.
P1 Establishing $2\times 3^{4}\times 5^{4}$ or an equivalent valid method.
A1 Correct answer: $101250$

Integer powers and roots

Type your working online and see every mark you earned and lost.