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 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
Between which two consecutive integers does 30\sqrt{30} lie?

Worked solution

  1. Find the square numbers either side of 30

    25<30<3625 < 30 < 36

    30 is trapped between 25, which is 5 squared, and 36, which is 6 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 30 lies between 5 and 6.

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 80

    81=9281 = 9^2

    80 sits right next to the square number 81.

  2. Compare the distances

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

    80 is only 1 away from 81, but 16 away from the square below, 64.

  3. State the closest integer

    80≈9\sqrt{80} \approx 9

    Since 80 is almost exactly 81, root 80 is very close to 9.

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 20

    8<20<278 < 20 < 27

    List cube numbers: 8 is 2 cubed and 27 is 3 cubed, so 20 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 20 lies between 2 and 3.

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 26

    25=5225 = 5^2

    26 sits right next to the square number 25.

  2. Compare the distances

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

    26 is only 1 above 25, but 10 below the next square, 36.

  3. State the closest integer

    26≈5\sqrt{26} \approx 5

    So root 26 is closest to 5.

Answer
55

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