Stage 9 - Q1 - 2 marks
Put the root between consecutive integers.
Hint
Find the square just below the radicand.
Stage 9 - Q2 - 2 marks
Put the root between consecutive integers.
Hint
Compare with nearby cube numbers.
Stage 9 - Q3 - 3 marks
Round the root to the nearest integer.
Hint
The halfway test is the square of the midpoint, not the midpoint of the squares.
Stage 9 - Q4 - 3 marks
Which is larger? Justify without a calculator.
Hint
Square both positive values.
Stage 9 - Q5 - 4 marks
A square has area 420 square units. Between which consecutive integers is its side length?
Hint
Write area as side squared.
Stage 9 - Q6 - 4 marks
A square has area 12 square units. Between which consecutive integers is its side length?
Hint
Write area as side squared.
Stage 9 - Q7 - 5 marks
Evaluate exactly without finding decimal roots.
Hint
Expand the two brackets and collect the middle terms.
Stage 9 - Q8 - 5 marks
Find all positive integers satisfying the inequality.
Hint
Square the endpoints, then use the integer condition.
Stage 9 - Q9 - 5 marks
A learner estimates the square root of 90 as 10 because 90 is near the middle of two squares. Is the nearest-integer answer correct?
Hint
Compare with the square of the halfway root.
Stage 9 - Q10 - 3 marks
Find all positive integers \(n\) for which the square root of \(n\) rounds to 7, and the cube root of \(n\) lies strictly between 3 and 4.
Hint
Convert both root conditions into intervals for \(n\).