Stage 8 - Q1 - 2 marks
Evaluate.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q2 - 2 marks
Evaluate.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q3 - 3 marks
Find every real solution.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q4 - 3 marks
Find every real solution.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q5 - 4 marks
A cube has volume 8 cubic centimetres. Find its side length and the total length of its 12 edges.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q6 - 4 marks
A cube has volume 27 cubic centimetres. Find its side length and the total length of its 12 edges.
Hint
Think about which number gives the stated square or cube. Check both signs when solving a square equation.
Stage 8 - Q7 - 5 marks
Find every positive integer n for which \(80<n^{2}<150\).
Hint
Find the smallest and largest square numbers strictly inside the interval.
Stage 8 - Q8 - 5 marks
A pupil claims that \(\sqrt{a+b}=\sqrt{a}+\sqrt{b}\) for all positive a, b. Use \(a=36\) and \(b=64\) to test the claim.
Hint
Do not simplify the roots with the proposed rule; evaluate each side independently.
Stage 8 - Q9 - 5 marks
An integer n has square 64 and a negative cube. Find n and the value of \(2n^{3}-n\).
Hint
Which of the two square roots has a negative cube?
Stage 8 - Q10 - 5 marks
A cube has an integer side length from 2 to 20 cm inclusive. Its volume is also a square number of cubic centimetres. Find every possible side length and volume, and justify completeness.
Hint
In a square number, prime factors occur in pairs. What happens to each prime count when the side is cubed?