Hard GCSE Estimating powers and roots Questions

Challenging, exam-style GCSE Estimating powers and roots questions with worked solutions. Stretch yourself on the hardest trial squares, one decimal place, two decimal places, proof problems.

trial squaresone decimal placetwo decimal placesproofnearest squarenearest cube
GCSE Higher34 questionsStep-by-step solutions
Question 1
6 markschallenging
Find the smallest integer nn such that n>20.5\sqrt{n} > 20.5.
Show worked solution

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
Question 2
5 markschallenging
Given that 992=980199^2 = 9801, write down the integer closest to 9750\sqrt{9750}. You must show your working.
Show worked solution

Worked solution

  1. State the known fact

    992=980199^2 = 9801

    The given square is close to 9750, so root 9750 is close to 99.

  2. Compare with the target

    9750<98019750 < 9801

    9750 is below 99 squared, so its root is below 99.

  3. Find the square below

    982=960498^2 = 9604

    98 squared is 9604, below 9750.

  4. Bracket the root

    98<9750<9998 < \sqrt{9750} < 99

    The root is trapped between 98 and 99.

  5. Plan the midpoint test

    test 98.5\text{test } 98.5

    Compare the root with 98.5 to decide the nearest integer.

  6. Square the midpoint

    98.52=9702.2598.5^2 = 9702.25

    98.5 squared is 9702.25.

  7. Compare with 9750

    9702.25<97509702.25 < 9750

    9750 is beyond the midpoint square.

  8. Interpret

    9750>98.5\sqrt{9750} > 98.5

    The root is in the upper half of the gap — closer to 99.

  9. Refine with 98.7

    98.72=9741.6998.7^2 = 9741.69

    98.7 squared is 9741.69, still below 9750 — the root is above 98.7.

  10. Refine with 98.8

    98.82=9761.4498.8^2 = 9761.44

    98.8 squared is 9761.44, above 9750 — the root is below 98.8.

  11. Locate the root

    98.7<9750<98.898.7 < \sqrt{9750} < 98.8

    So the root is about 98.7 or so — clearly nearer 99 than 98.

  12. Use the gap sizes as a check

    9750−9604=146,9801−9750=519750 - 9604 = 146,\quad 9801 - 9750 = 51

    9750 hugs the 9801 end of the bracket, agreeing that the root is close to 99.

  13. Round to the nearest integer

    9750≈99\sqrt{9750} \approx 99

    Being above 98.5, the root rounds to 99.

  14. Sense check

    9750≈98.74\sqrt{9750} \approx 98.74

    The true value is about 98.74, which indeed rounds to 99. ✓

  15. State the answer

    9999

    The integer closest to root 9750 is 99.

Answer
9999
Question 3
6 markschallenging
x4=50x^4 = 50 and x>0x > 0. Between which two consecutive one-decimal-place numbers does xx lie? You must show your working.
Show worked solution

Worked solution

  1. Understand the equation

    x4=50,x>0x^4 = 50,\quad x > 0

    We need the positive fourth root of 50, bracketed between consecutive tenths.

  2. Plan via repeated squaring

    x4=(x2)2x^4 = (x^2)^2

    A fourth power is a square of a square, so trial values can be squared twice.

  3. Bracket with whole numbers

    24=16,34=812^4 = 16,\quad 3^4 = 81

    50 sits between 16 and 81, so x is between 2 and 3.

  4. Judge where in the gap

    50≈7.07\sqrt{50} \approx 7.07

    Since x squared is root 50, about 7.07, x is around root 7.07 — roughly 2.66. Start trials near there.

  5. Try 2.6: square once

    2.62=6.762.6^2 = 6.76

    First squaring of 2.6.

  6. Square again

    6.762=45.69766.76^2 = 45.6976

    2.6 to the fourth is 45.6976, below 50 — so x is above 2.6.

  7. Try 2.7: square once

    2.72=7.292.7^2 = 7.29

    First squaring of 2.7.

  8. Square again

    7.292=53.14417.29^2 = 53.1441

    2.7 to the fourth is 53.1441, above 50 — so x is below 2.7.

  9. State the bracket

    2.6<x<2.72.6 < x < 2.7

    x is trapped between the consecutive tenths 2.6 and 2.7.

  10. Verify the squares used

    6762=456976676^2 = 456976

    Checking: 676 squared is 456976, so 6.76 squared is 45.6976 as claimed.

  11. Verify the other square

    7292=531441729^2 = 531441

    729 squared is 531441, so 7.29 squared is 53.1441 as claimed.

  12. Sense check

    x=5014≈2.659x = 50^{\frac{1}{4}} \approx 2.659

    The true fourth root of 50 is about 2.66, inside the bracket. ✓

  13. Connect to fractional indices

    x=5014=50x = 50^{\frac{1}{4}} = \sqrt{\sqrt{50}}

    The fourth root is the square root taken twice — the same structure as the repeated squaring used in the trials.

  14. Reflect on the method

    square twice instead of multiplying four times\text{square twice instead of multiplying four times}

    Squaring twice keeps each trial to two easy multiplications.

  15. State the answer

    2.6 and 2.72.6 \text{ and } 2.7

    x lies between 2.6 and 2.7.

Answer
2.6 and 2.72.6 \text{ and } 2.7
Question 4
5 markschallenging
A student estimates 60\sqrt{60} as 36+1002=6+102=8\frac{\sqrt{36} + \sqrt{100}}{2} = \frac{6+10}{2} = 8. Explain why this method is wrong and find a better estimate of 60\sqrt{60} to one decimal place.
Show worked solution

Worked solution

  1. State the student's method

    60≈36+1002=8\sqrt{60} \approx \frac{\sqrt{36}+\sqrt{100}}{2} = 8

    The student averaged the roots of 36 and 100 because 60 lies between them.

  2. Test the answer

    82=64≠608^2 = 64 \ne 60

    If 8 were close to root 60, then 8 squared should be close to 60 — but it is 64, noticeably off.

  3. Identify the first flaw

    60≠36+1002=6860 \ne \frac{36+100}{2} = 68

    60 is not midway between 36 and 100 — the midpoint of those squares is 68, so the average is anchored to the wrong place.

  4. Identify the deeper flaw

    a+b2≠a+b2\sqrt{\tfrac{a+b}{2}} \ne \tfrac{\sqrt{a}+\sqrt{b}}{2}

    Even at the exact midpoint, averaging roots overestimates: the square root graph curves, so the straight-line average sits above it.

  5. See the flaw numerically

    68≈8.25>8\sqrt{68} \approx 8.25 > 8

    Root 68 is about 8.25 — and root 60 must be smaller still, so 8 cannot be right for root 60.

  6. Start the correct method

    49<60<6449 < 60 < 64

    Bracket 60 with the nearest squares: 7 squared and 8 squared.

  7. Bracket the root

    7<60<87 < \sqrt{60} < 8

    Root 60 is between 7 and 8 — already the student's 8 is exposed as an endpoint, not an estimate.

  8. Try 7.7

    7.72=59.297.7^2 = 59.29

    7.7 squared is 59.29, just below 60.

  9. Try 7.8

    7.82=60.847.8^2 = 60.84

    7.8 squared is 60.84, just above 60.

  10. Bracket to tenths

    7.7<60<7.87.7 < \sqrt{60} < 7.8

    Root 60 is trapped between 7.7 and 7.8.

  11. Test the halfway value

    7.752=60.06257.75^2 = 60.0625

    7.75 squared is just above 60, so root 60 is just below 7.75.

  12. Round to one decimal place

    60≈7.7\sqrt{60} \approx 7.7

    Being below 7.75, root 60 rounds to 7.7.

  13. Compare the two estimates

    7.7 vs 87.7 \text{ vs } 8

    The trial-squares estimate of 7.7 is far more accurate than the student's 8 — the true value is about 7.746.

  14. Summarise the error

    roots cannot be averaged\text{roots cannot be averaged}

    In one sentence: averaging the roots of two convenient squares is not a valid way to estimate a root in between.

  15. State the better estimate

    60≈7.7\sqrt{60} \approx 7.7

    A better estimate of root 60 is 7.7 to one decimal place.

Answer
60≈7.7\sqrt{60} \approx 7.7
Question 5
6 markschallenging
Given that 412=168141^2 = 1681, estimate 170000\sqrt{170000} to the nearest integer. You must show your working.
Show worked solution

Worked solution

  1. State the known fact

    412=168141^2 = 1681

    We must stretch this single fact to estimate the root of 170000.

  2. Scale the fact by 100

    4102=168100410^2 = 168100

    Multiplying 41 by 10 multiplies its square by 100, giving a square very close to 170000.

  3. Compare with the target

    168100<170000168100 < 170000

    170000 is a little beyond 410 squared, so its root is a little beyond 410.

  4. Gauge the gap

    170000−168100=1900170000 - 168100 = 1900

    The target is 1900 above the known square.

  5. Estimate the step per unit

    4112−4102=821411^2 - 410^2 = 821

    Each step of 1 in the root adds about 820 to the square at this size (the two roots sum to 821).

  6. Estimate the extra distance

    1900÷821≈2.31900 \div 821 \approx 2.3

    1900 divided by roughly 820 suggests the root is about 2.3 above 410.

  7. Form the candidate

    170000≈412.3\sqrt{170000} \approx 412.3

    So the root should be near 412, and we check the neighbouring integers.

  8. Check 412

    4122=169744412^2 = 169744

    412 squared is 169744, below 170000.

  9. Check 413

    4132=170569413^2 = 170569

    413 squared is 170569, above 170000.

  10. Bracket the root

    412<170000<413412 < \sqrt{170000} < 413

    The root is trapped between 412 and 413.

  11. Test the midpoint

    412.52=170156.25412.5^2 = 170156.25

    The midpoint squares to just above 170000.

  12. Interpret the test

    170000<412.5\sqrt{170000} < 412.5

    The root is below 412.5, so it rounds down to 412.

  13. Round to the nearest integer

    170000≈412\sqrt{170000} \approx 412

    To the nearest integer, the root is 412.

  14. Sense check

    170000≈412.31\sqrt{170000} \approx 412.31

    The true value is about 412.3 — matching the step-size estimate from earlier. ✓

  15. State the answer

    412412

    The nearest integer to root 170000 is 412.

Answer
412412

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