Estimating powers and roots Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Estimating powers and roots questions. See exactly how to solve problems on bracketing square roots, square numbers, nearest integer, bracketing cube roots.

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?

Worked solution

  1. Find the square numbers either side of 5050

    49<50<6449 < 50 < 64

    List square numbers until 5050 is trapped: 4949 is 77 squared and 6464 is 88 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 5050 stays trapped in the middle.

  3. State the integers

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

    So root 5050 lies between the consecutive integers 77 and 88.

Answer
7 and 87 \text{ and } 8
Question 2
2 markseasy
Between which two consecutive integers does 30\sqrt{30} lie?

Worked solution

  1. Find the square numbers either side of 3030

    25<30<3625 < 30 < 36

    3030 is trapped between 2525, which is 55 squared, and 3636, which is 66 squared.

  2. Take square roots across the bracket

    25<30<36\sqrt{25} < \sqrt{30} < \sqrt{36}

    Rooting each part keeps the order.

  3. State the integers

    5<30<65 < \sqrt{30} < 6

    So root 3030 lies between 55 and 66.

Answer
5 and 65 \text{ and } 6
Question 3
1 markeasy
Write down the integer closest to 80\sqrt{80}.

Worked solution

  1. Look for a square number close to 8080

    81=9281 = 9^2

    8080 sits right next to the square number 8181.

  2. Compare the distances

    80−64=16,81−80=180 - 64 = 16,\quad 81 - 80 = 1

    8080 is only 11 away from 8181, but 1616 away from the square below, 6464.

  3. State the closest integer

    80≈9\sqrt{80} \approx 9

    Since 8080 is almost exactly 8181, root 8080 is very close to 99.

Answer
99
Question 4
2 markseasy
Between which two consecutive integers does 203\sqrt[3]{20} lie?

Worked solution

  1. Find the cube numbers either side of 2020

    8<20<278 < 20 < 27

    List cube numbers: 88 is 22 cubed and 2727 is 33 cubed, so 2020 is trapped between them.

  2. Take cube roots across the bracket

    83<203<273\sqrt[3]{8} < \sqrt[3]{20} < \sqrt[3]{27}

    Cube rooting keeps the order.

  3. State the integers

    2<203<32 < \sqrt[3]{20} < 3

    So the cube root of 2020 lies between 22 and 33.

Answer
2 and 32 \text{ and } 3
Question 5
1 markeasy
Write down the integer closest to 26\sqrt{26}.

Worked solution

  1. Look for a square number close to 2626

    25=5225 = 5^2

    2626 sits right next to the square number 2525.

  2. Compare the distances

    26−25=1,36−26=1026 - 25 = 1,\quad 36 - 26 = 10

    2626 is only 11 above 2525, but 1010 below the next square, 3636.

  3. State the closest integer

    26≈5\sqrt{26} \approx 5

    So root 2626 is closest to 55.

Answer
55

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