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Cambridge IGCSE0580

Powers and roots

Mathematics 0580 Chapter Notes

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Powers and roots
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In this chapter

  1. 1Squares and square roots
  2. 2Cubes and cube roots
  3. 3Other powers
  4. 4Powers and roots of decimals
  5. 5Powers and roots on a calculator
  6. 6Powers in a longer calculation

1. Squares and square roots

A square is a number multiplied by itself: n² = n × n. A square root undoes it, so √(169) = 13 because 13² = 169.

Learn the squares from 1 to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Each square gives a root: √81 = 9.

n² = n × n

√n² = n

Examiner insight

“Some read ‘square’ as ‘square root’.”

Worked example 12 marks

K = 2x + √y

Find the value of K when x = 5 and y = 169.

  1. 1

    √(169) = 13 [1]

  2. 2

    K = 2 × 5 + 13 = 23 [1]

2. Cubes and cube roots

A cube is three of the same number multiplied together: n³ = n × n × n. A cube root undoes it, so √[3]27 = 3 because 3³ = 27.

Learn the cubes of 1, 2, 3, 4, 5 and 10: 1, 8, 27, 64, 125, 1000. 9 is a square number, not a cube number. The small 3 in √[3]40 asks for a cube root, not 3 times a square root.

To work out 5 squared times the cube root of 8, find each part first. 5² = 25 and √[3]8 = 2, so the answer is 25 × 2 = 50.

n³ = n × n × n

√[3]n³ = n

Examiner insight

“While many did know how to find the cube root using their calculators, it was common to see three times the square root of 40.” “A few candidates thought 9 was a cube number.”

Worked example 11 mark

(a) Write down the value of √[3]27.

  1. 1

    3 × 3 × 3 = 27

  2. 2

    √[3]27 = 3 [1]

3. Other powers

The index says how many of the number are multiplied together. 5² = 5 × 5 and 2³ = 2 × 2 × 2. Six 7s multiplied together, 7 × 7 × 7 × 7 × 7 × 7, is written 7⁶.

An index is not a multiplier: 5² is not 5 × 2. Work out each power first. 5² = 25 and 2³ = 8, so their difference is 17.

Examiner insight

“Those not understanding indices usually gave responses of 10 and 6 leading to an answer of 4.”

Worked example 12 marks

Work out the difference between 5² and 2³.

  1. 1

    5² = 25 and 2³ = 8 [1]

  2. 2

    25 - 8 = 17 [1]

4. Powers and roots of decimals

To square a decimal, multiply the digits as whole numbers, then count decimal places: 0.7 has one, so 0.7² has two. 7 × 7 = 49, so 0.7² = 0.49.

To take the root of a decimal, write it as a fraction. 0.0025 = 25/10000, so the root is 5/100 = 0.05.

0.7² = 0.49

√(0.0025) = 0.05

Examiner insight

“Some completed the multiplication correctly then inserted the decimal point in the incorrect place giving answers such as 4.9 or 0.049.”

Worked example 11 mark

Work out √(0.0025).

  1. 1

    0.0025 = 25/10000

  2. 2

    √25 = 5 and √(10000) = 100

  3. 3

    √(0.0025) = 5/100 = 0.05 [1]

5. Powers and roots on a calculator

Papers 3 and 4 allow a calculator. Use its power key for a power such as 17.4² = 302.76 and its root keys for a root such as √[3]1.728 = 1.2.

Write all the figures the calculator shows before you round. Bracket everything under a root sign. Bracket the whole bottom of a fraction too.

Examiner insight

“While many candidates gave a correct answer there were a significant number who did not follow the instruction to include all the numbers on their calculator, and specifically gave less than the minimum of seven figures required for the mark.” “A common incorrect answer of –72.997125 was the result of not bracketing or working out the calculation under the cube root.”

Worked example 12 marks

Calculate.

(a) √[3]1.728

(b) 18/(0.2² - 0.36)

  1. 1

    (a) √[3]1.728 = 1.2 [1]

  2. 2

    (b) bottom of the fraction: 0.2² - 0.36 = 0.04 - 0.36 = -0.32

  3. 3

    18 ÷ (-0.32) = -56.25 [1]

6. Powers in a longer calculation

Powers and roots come before multiplication. Multiplication comes before subtraction.

In 10 - (7/8)³ × 2, cube first, then multiply by 2, and subtract from 10 last.

Keep the full calculator value and round only at the end. 8.66015625 to 1 decimal place is 8.7.

Examiner insight

“Some candidates incorrectly rounded 8.660 to 8.6 or rounded to 8.66.”

Worked example 12 marks

Calculate 10 - (7/8)³ × 2.

Give your answer correct to 1 decimal place.

  1. 1

    (7/8)³ × 2 = 1.33984375, worked out before subtracting [1]

  2. 2

    10 - 1.33984375 = 8.66015625, which is 8.7 to 1 decimal place [1]

Recap

  • Square: n² = n × n. Learn the squares 1 to 15 and their roots.
  • Cube: n³ = n × n × n. Learn the cubes of 1, 2, 3, 4, 5 and 10 and their roots.
  • The index says how many of the number are multiplied together; it is not a multiplier.
  • Squaring a decimal doubles the number of decimal places: 0.7² = 0.49.
  • On a calculator, bracket everything under a root sign and the whole bottom of a fraction.
  • Powers first, then multiplication, then subtraction. Round only at the end.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the value of √(144).1 mark
  2. Write down the cube of 5.1 mark
  3. Do not use a calculator. Work out √49 + 3².2 marks
  4. Do not use a calculator. Work out 6 squared minus the cube root of 27.2 marks
  5. Do not use a calculator. (a) Work out 0.3² + 0.4². (b) Write down the square root of your answer to part (a).3 marks
  6. Do not use a calculator. Work out √(0.0016).1 mark
  7. Calculate √[3]216 + 2.5⁴. Give your answer correct to 2 decimal places.2 marks
  8. Calculate 12 - 3 × 1.2³. Give your answer correct to 1 decimal place.3 marks
  9. Do not use a calculator. Amira thinks of a positive whole number. She squares it, then takes the cube root of the result. She gets 4. Work out the cube of her number.3 marks
  10. The volume of a cube is the cube of its edge length. A metal cube of edge 4.5 cm and a metal cube of edge 5.2 cm are melted and made into one new cube. No metal is lost. (a) Calculate the edge length of the new cube. Give your answer correct to 2 decimal places. (b) Calculate the total area of the six square faces of the new cube. Give your answer correct to 1 decimal place.5 marks

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