Stage 8 - Q1 - 2 marks
Evaluate.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q2 - 2 marks
Evaluate.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q3 - 3 marks
Calculate.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q4 - 3 marks
Calculate.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q5 - 4 marks
A pattern starts with 7 tiles at stage 0 and doubles at each stage. Find the number at stage 2. Write a calculation using a power.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q6 - 4 marks
A pattern starts with 6 tiles at stage 0 and doubles at each stage. Find the number at stage 4. Write a calculation using a power.
Hint
Expand a power into repeated factors or work backwards from the final number.
Stage 8 - Q7 - 5 marks
A display doubles each minute. After five doublings it has 96 tiles. How many tiles did it have initially?
Hint
Write the total multiplier for five doublings.
Stage 8 - Q8 - 5 marks
Which is larger: \(3^{4}\) or \(4^{3}\)? Explain why swapping a base and an index does not preserve the value.
Hint
Expand both powers as repeated multiplication.
Stage 8 - Q9 - 5 marks
Find every positive integer n for which \(2^{n}<100\) and \(3^{n}>20\).
Hint
Find the boundary exponent for each inequality, then overlap the possibilities.
Stage 8 - Q10 - 5 marks
Find every pair of positive integers \(a,b\) with \(a>1\) and \(b>1\) such that \(a^b=64\).
Hint
Use the prime factorisation of 64 to restrict the possible bases.