GCSE Estimating powers and roots Practice Questions

Free GCSE Estimating powers and roots practice questions with full step-by-step worked solutions. Covers bracketing square roots, square numbers, nearest integer, bracketing cube roots. Practise exam-style problems and check your method.

bracketing square rootssquare numbersnearest integerbracketing cube rootscube numbersestimating powers
GCSE Higher70 questionsStep-by-step solutions
Question 1
2 markseasy
Between which two consecutive integers does 50\sqrt{50} lie?
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Worked solution

  1. Find the square numbers either side of 50

    49<50<6449 < 50 < 64

    List square numbers until 50 is trapped: 49 is 7 squared and 64 is 8 squared.

  2. Take square roots across the bracket

    49<50<64\sqrt{49} < \sqrt{50} < \sqrt{64}

    Square rooting keeps the order for positive numbers, so root 50 stays trapped in the middle.

  3. State the integers

    7<50<87 < \sqrt{50} < 8

    So root 50 lies between the consecutive integers 7 and 8.

Answer
7 and 87 \text{ and } 8
Question 2
2 markseasy
Which is greater: 60\sqrt{60} or 88? You must show your working.
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Worked solution

  1. Square the whole number

    82=648^2 = 64

    To compare a root with a whole number, square them both. 8 squared is 64.

  2. Square the root

    (60)2=60(\sqrt{60})^2 = 60

    Squaring root 60 gives exactly 60.

  3. Compare the squares

    64>60 ⇒ 8>6064 > 60 \ \Rightarrow\ 8 > \sqrt{60}

    Since 64 beats 60, the number 8 is greater than root 60.

Answer
8 is greater, because 82=64>608 \text{ is greater, because } 8^2 = 64 > 60
Question 3
2 marksintermediate
Which is greater: 653\sqrt[3]{65} or 17\sqrt{17}? You must show your working.
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Worked solution

  1. Pick a common test value

    test 4.1\text{test } 4.1

    Both roots are a little above 4, so test each against the same value, 4.1.

  2. Test the square root

    4.12=16.81<174.1^2 = 16.81 < 17

    4.1 squared is 16.81, below 17 — so root 17 is above 4.1.

  3. Test the cube root

    4.13=68.921>654.1^3 = 68.921 > 65

    4.1 cubed is 68.921, above 65 — so the cube root of 65 is below 4.1.

  4. Put the results together

    653<4.1<17\sqrt[3]{65} < 4.1 < \sqrt{17}

    The same test value 4.1 sits between the two roots.

  5. Draw the conclusion

    17>653\sqrt{17} > \sqrt[3]{65}

    So root 17 is the greater number.

  6. Sense check with decimals

    17≈4.12,653≈4.02\sqrt{17} \approx 4.12,\quad \sqrt[3]{65} \approx 4.02

    The approximate values 4.12 and 4.02 confirm the comparison.

Answer
17 is greater, since 653<4.1<17\sqrt{17} \text{ is greater, since } \sqrt[3]{65} < 4.1 < \sqrt{17}
Question 4
4 markshard
Write these numbers in order of size, starting with the smallest: 99\sqrt{99}, 9.99.9, 9803\sqrt[3]{980}.
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Worked solution

  1. Plan the comparison

    99, 9.9, 9803\sqrt{99},\ 9.9,\ \sqrt[3]{980}

    All three values are close to 9.9, so compare each root against carefully chosen decimals.

  2. Compare root 99 with 9.9

    9.92=98.01<999.9^2 = 98.01 < 99

    9.9 squared falls short of 99, so root 99 is bigger than 9.9.

  3. Compare the cube root with 9.9

    9.93=970.299<9809.9^3 = 970.299 < 980

    9.9 cubed falls short of 980, so the cube root of 980 is also bigger than 9.9. So 9.9 is the smallest.

  4. Now separate the two roots

    test 9.94\text{test } 9.94

    Both roots beat 9.9, so test a finer value against each.

  5. Test the cube root at 9.94

    9.943≈982.1>9809.94^3 \approx 982.1 > 980

    9.94 cubed is about 982, above 980 — so the cube root of 980 is below 9.94.

  6. Test root 99 at 9.94

    9.942=98.8036<999.94^2 = 98.8036 < 99

    9.94 squared is 98.80, below 99 — so root 99 is above 9.94.

  7. Put the roots in order

    9803<9.94<99\sqrt[3]{980} < 9.94 < \sqrt{99}

    The same test value 9.94 separates the two roots.

  8. Assemble the full order

    9.9<9803<999.9 < \sqrt[3]{980} < \sqrt{99}

    Combining both comparisons gives the complete ordering.

  9. Sense check with decimals

    9803≈9.933,99≈9.950\sqrt[3]{980} \approx 9.933,\quad \sqrt{99} \approx 9.950

    The approximate values confirm the order: 9.9, then 9.933, then 9.950.

  10. State the answer

    9.9, 9803, 999.9,\ \sqrt[3]{980},\ \sqrt{99}

    From smallest to largest: 9.9, the cube root of 980, then root 99.

Answer
9.9, 9803, 999.9,\ \sqrt[3]{980},\ \sqrt{99}
Question 5
6 markschallenging
Find the smallest integer nn such that n>20.5\sqrt{n} > 20.5.
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Worked solution

  1. Understand the condition

    n>20.5\sqrt{n} > 20.5

    We need the smallest integer n whose square root beats 20.5.

  2. Square both sides

    n>20.52n > 20.5^2

    Squaring keeps the order for positive numbers, turning the root condition into a plain inequality.

  3. Plan the squaring

    20.52=(20+0.5)220.5^2 = (20 + 0.5)^2

    Split 20.5 to square it mentally.

  4. Expand the square

    202+2×20×0.5+0.5220^2 + 2 \times 20 \times 0.5 + 0.5^2

    Use the expansion of a two-part square: 400 plus 20 plus 0.25.

  5. Evaluate

    20.52=420.2520.5^2 = 420.25

    So the condition is n greater than 420.25.

  6. Find the smallest integer

    nmin⁡=421n_{\min} = 421

    The smallest integer strictly greater than 420.25 is 421.

  7. Check 421 works

    421>420.25=20.5\sqrt{421} > \sqrt{420.25} = 20.5

    421 is beyond 420.25, so its root is beyond 20.5. ✓

  8. Check 420 fails

    420<420.25 ⇒ 420<20.5420 < 420.25 \ \Rightarrow\ \sqrt{420} < 20.5

    420 falls short of 420.25, so its root falls short of 20.5 — 420 fails.

  9. Confirm minimality

    420 fails, 421 works420 \text{ fails},\ 421 \text{ works}

    Since 420 fails and 421 works, 421 is the smallest possible n.

  10. Estimate the root of 421 as a check

    20.52=420.25, 20.522≈421.0720.5^2 = 420.25,\ 20.52^2 \approx 421.07

    Root 421 is about 20.518, just above 20.5 — a comfortable but tight pass.

  11. Watch for the boundary error

    n=420 would need 420>20.5n = 420 \text{ would need } \sqrt{420} > 20.5

    Rounding 420.25 down to 420 is the classic slip; the strict inequality demands going above 420.25.

  12. Watch for the equality trap

    420.25=20.5≯20.5\sqrt{420.25} = 20.5 \not> 20.5

    Even the exact value 420.25 (if n could be a decimal) only equals 20.5 — the strict inequality rules it out.

  13. Reflect on the method

    square the threshold, step up\text{square the threshold, step up}

    Threshold problems with roots are solved by squaring the boundary and taking the next integer.

  14. General lesson

    n>k  ⟺  n>k2\sqrt{n} > k \iff n > k^2

    For positive values, root conditions convert exactly into squared conditions.

  15. State the answer

    n=421n = 421

    The smallest integer n is 421.

Answer
n=421n = 421

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