Stage 8 - Q1 - 2 marks
Write as one power.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q2 - 2 marks
Write as one power.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q3 - 3 marks
Write as one power.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q4 - 3 marks
Write as one power.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q5 - 4 marks
Simplify to a single power. Explain the two laws used.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q6 - 4 marks
Simplify to a single power. Explain the two laws used.
Hint
Rewrite everything using one base, then combine the indices.
Stage 8 - Q7 - 5 marks
Without evaluating the large numbers, decide whether \(2^{12}\) and \(4^{6}\) are equal. Explain.
Hint
Express both powers using base 2.
Stage 8 - Q8 - 5 marks
Find the positive integer n.
Hint
Place brackets around the entire index being subtracted.
Stage 8 - Q9 - 5 marks
Positive integers m and n satisfy \(2^m\times 2^n=256\) and \((2^m)^n=32768\). Find m and n.
Hint
One index law gives the sum of m, n and the other gives their product.
Stage 8 - Q10 - 5 marks
Find all positive integers \(m,n\) with \(m>n\) for which \(2^m\times4^n=1024\).
Hint
Rewrite 4 and 1024 as powers of 2 before comparing exponents.