Mathematics notes, chapter 3

Squares, Square Roots, Cubes and Cube Roots

Textbook Mathematics Form 1 (DLP), pages 46–72

What this chapter covers

Explain the meaning of squares and perfect squares, and decide whether a number is a perfect square by prime factorisation (3.1)
State the relationship between squares and square roots, and find squares and square roots with and without a calculator (3.1)
Estimate squares and square roots, and multiply square roots (√a × √a = a, √a × √b = √ab) (3.1)
Solve problems involving squares and square roots (3.1)
Explain the meaning of cubes and perfect cubes, and decide whether a number is a perfect cube by prime factorisation (3.2)
State the relationship between cubes and cube roots, and find cubes and cube roots with and without a calculator (3.2)
Estimate cubes and cube roots, and solve problems involving cubes and cube roots (3.2)
Work out combined operations involving squares, square roots, cubes and cube roots (3.2)

At the 2015 SEA Games our silat team competed on a square court with an area of 100 m². How long is each side? We need a number that, multiplied by itself, gives 100. That number is 10, so each side is 10 m. In this chapter you will learn about squares and square roots (for areas of squares) and cubes and cube roots (for volumes of cubes).

English	Bahasa Melayu
square, perfect square	kuasa dua, kuasa dua sempurna
square root	punca kuasa dua
cube, perfect cube	kuasa tiga, kuasa tiga sempurna
cube root	punca kuasa tiga
estimation	anggaran

3.1Squares and square roots

Squares and perfect squares

The square of a number is the number multiplied by itself. The square of 4 is 4 × 4 = 16. We write 4² = 16 and read 4² as "four squared" or "the square of four".

A square with sides of 4 units has an area of 4 × 4 = 16 unit². That is why we call it "squaring".

Squares of side 1, 2, 3 and 4 units made of 1, 4, 9 and 16 unit squares — The area of a square is the square of its side length

A perfect square is a number that is the square of a whole number: 1, 4, 9, 16, 25, … That many 1 unit × 1 unit tiles can be arranged into a square, with nothing left over.

Number	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Square	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Learn the perfect squares up to 15² = 225 by heart. They make square roots and estimation much faster.

4² means 4 × 4 = 16, not 4 × 2 = 8. The small 2 tells you how many times the number is multiplied by itself.

Is the number a perfect square?

Use prime factorisation. If the prime factors can be split into two identical groups, the number is a perfect square.

Example 1: 36 and 54 (textbook p. 50)

Factor tree: 36 = 4 × 9, 4 = 2 × 2, 9 = 3 × 3

Factor tree: 54 = 6 × 9, 6 = 2 × 3, 9 = 3 × 3

36 = 2 × 2 × 3 × 3 = (2 × 3) × (2 × 3) = 6 × 6. Two identical groups, so 36 is a perfect square.
54 = 2 × 3 × 3 × 3. There is only one 2, so the factors cannot be split into two identical groups. 54 is not a perfect square.

Use prime factorisation. Which are perfect squares? a) 45 b) 100 c) 324

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a) 45 = 3 × 3 × 5. Only one 5, so not a perfect square.
b) 100 = 2 × 2 × 5 × 5 = (2 × 5) × (2 × 5) = 10 × 10. Perfect square.
c) 324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 3 × 3) × (2 × 3 × 3) = 18 × 18. Perfect square.

Squares and square roots are inverse operations

The square root of a number is the number that was squared to make it. 6² = 36, so the square root of 36 is 6. We write √36 = 6 and read it as "square root of thirty-six".

6 squared gives 36; the square root of 36 gives 6 — Squaring and finding the square root undo each other

Finding the square root of the area of a square gives the length of its side. A square with an area of 36 unit² has sides of √36 = 6 units.

(−5) × (−5) = 25 too, so 25 has two square roots, 5 and −5. The symbol √ always means the positive one: √25 = 5.

Finding the square of a number

Multiply the number by itself. This works for whole numbers, fractions, decimals and negative numbers.

Example 2: without a calculator (textbook p. 52)

6² = 6 × 6 = 36
(34)² = 34 × 34 = 916
(−0.5)² = (−0.5) × (−0.5) = 0.25 (negative × negative = positive)

Example 3: with a calculator (textbook p. 52)

Question	Press	Answer
43²	4 3 x² =	1 849
(−713)²	( (−) 7 a b/c 1 3 ) x² =	49169
2.96²	2 . 9 6 x² =	8.7616

The square of a negative number is always positive: (−6)² = 36. On a calculator, put the negative number in brackets, ( (−) 6 ) x², or you will get −36.

0.3² = 0.3 × 0.3 = 0.09, not 0.9. Count the decimal places: 1 + 1 = 2 decimal places.

Without a calculator: a) 15² b) 0.3² c) (−112)²

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a) 15 × 15 = 225
b) 0.3 × 0.3 = 0.09
c) (−32) × (−32) = 94 = 214

Finding square roots without a calculator

For a small perfect square, recall the times table. For a bigger number, use prime factorisation and split the factors into two identical groups. One group is the square root.

Example 4: √441 (textbook p. 53)

Factor tree: 441 = 9 × 49, 9 = 3 × 3, 49 = 7 × 7

441 = 3 × 3 × 7 × 7
= (3 × 7) × (3 × 7) = 21 × 21
√441 = 21

Example 5: fractions and decimals (textbook pp. 53–54)

√425 = √4√25 = 25
√279 = √259 = 53 = 123 (change to an improper fraction first)
√2748 = √916 = 34 (simplify the fraction first)
√0.36 = √36100 = 610 = 0.6

√279 is not √2 and √79 put together. Always change a mixed number to an improper fraction first.

√0.09 = 0.3, not 0.03. Check by squaring: 0.3 × 0.3 = 0.09 ✓, but 0.03 × 0.03 = 0.0009.

Without a calculator: a) √0.09 b) √1916 c) √576 using prime factorisation

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a) √9100 = 310 = 0.3
b) √2516 = 54 = 114
c) 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = (2 × 2 × 2 × 3) × (2 × 2 × 2 × 3) = 24 × 24, so √576 = 24

Square roots with a calculator

Most numbers are not perfect squares, so their square roots are long decimals. Round them as asked.

Question	Press	Answer (2 d.p.)
√89	√ 8 9 =	9.43
√154.7	√ 1 5 4 . 7 =	12.44
√627	√ 6 a b/c 2 a b/c 7 =	2.51

Estimating squares and square roots

Find the two perfect squares on either side of the number, then choose the nearer one.

Example 6: estimate 27.5² and √54 (textbook p. 55)

27.5 is between 20 and 30, so 27.5² is between 20² = 400 and 30² = 900. 27.5 is nearer 30, so 27.5² ≈ 900.
54 is between the perfect squares 49 and 64, so √54 is between 7 and 8. 54 is nearer 49, so √54 ≈ 7.

Number line: 49 at 7, 64 at 8, 54 just past 7 at the square root of 54 — √54 lies between 7 and 8, closer to 7

a) Find √43 correct to 2 decimal places. b) Estimate √34.

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a) √43 = 6.557… = 6.56 (2 d.p.)
b) 34 is between 25 and 36, nearer 36, so √34 ≈ 6

Multiplying square roots

√a × √a = a (same numbers) √a × √b = √ab (different numbers)

Example 7

√7 × √7 = 7
√2 × √18 = √2 × 18 = √36 = 6

√2 × √18 is not √20. Multiply the numbers under the root sign, do not add them.

Solving problems

Example 8: Maslina's photo (textbook p. 56)

A square cardboard has sides of 12 cm. A square photo has an area of 90.25 cm². Where should Maslina paste the photo so that it is in the middle of the cardboard?

Square cardboard 12 cm with a square photo of area 90.25 cm² in the middle, 1.25 cm from each side

Side of photo = √90.25 = √9.5 × 9.5 = 9.5 cm
Length left over = 12 − 9.5 = 2.5 cm
Gap on each side = 2.5 ÷ 2 = 1.25 cm

Paste the photo 1.25 cm from each side. The area not covered is 12² − 90.25 = 144 − 90.25 = 53.75 cm².

3.2Cubes and cube roots

Cubes and perfect cubes

The cube of a number is the number multiplied by itself three times. The cube of 2 is 2 × 2 × 2 = 8. We write 2³ = 8 and read 2³ as "two cubed" or "the cube of two".

A cube with sides of 2 units is made of 2 × 2 × 2 = 8 unit cubes. Its volume is 8 unit³.

Cubes with sides 1, 2 and 3 units made of 1, 8 and 27 unit cubes — The volume of a cube is the cube of its side length

A perfect cube is the cube of a whole number: 1, 8, 27, 64, 125, … That many unit blocks can be built into a cube.

Number	1	2	3	4	5	6	7	8	9	10
Cube	1	8	27	64	125	216	343	512	729	1 000

Zaiton said "2³ = 2 × 3 = 6". This is wrong. 2³ = 2 × 2 × 2 = 8.

Is the number a perfect cube?

Use prime factorisation. If the prime factors can be split into three identical groups, the number is a perfect cube.

Example 9: 64 and 240 (textbook p. 60)

Factor tree: 64 = 8 × 8, each 8 = 2 × 4, each 4 = 2 × 2

Factor tree: 240 = 12 × 20, 12 = 3 × 4, 20 = 4 × 5, each 4 = 2 × 2

64 = 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2) × (2 × 2) × (2 × 2) = 4 × 4 × 4. 64 is a perfect cube.
240 = 2 × 2 × 2 × 2 × 3 × 5. There is only one 3 and one 5, so 240 is not a perfect cube.

64 is a perfect square and a perfect cube: 64 = 8² = 4³. Its six identical prime factors can be split into 2 groups or 3 groups.

Is 343 a perfect cube? Use prime factorisation.

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343 = 7 × 7 × 7. Three identical groups (one 7 each), so 343 is a perfect cube: 343 = 7³.

Cubes and cube roots are inverse operations

The cube root of a number is the number that was cubed to make it. 2³ = 8, so ³√8 = 2, read "cube root of eight".

2 cubed gives 8; the cube root of 8 gives 2

Finding the cube root of the volume of a cube gives the length of its side.

Signs: a cube of a positive number is positive, a cube of a negative number is negative. The same is true for cube roots: ³√−0.125 = −0.5 because (−0.5) × (−0.5) × (−0.5) = −0.125.

You cannot find the square root of a negative number (no number squared is negative), but you can find the cube root of a negative number.

Finding the cube of a number

Example 10: without a calculator (textbook p. 63)

4³ = 4 × 4 × 4 = 64
0.2³ = 0.2 × 0.2 × 0.2 = 0.008 (3 decimal places)
(−35)³ = (−35) × (−35) × (−35) = −27125

Example 11: with a calculator (textbook p. 63)

Question	Press	Answer
18³	1 8 x³ =	5 832
(−412)³	( (−) 4 a b/c 1 a b/c 2 ) x³ =	−9118
(−6.3)³	( (−) 6 . 3 ) x³ =	−250.047

a) (−7)³ b) ³√−125 c) ³√0.001

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a) (−7) × (−7) × (−7) = −343
b) (−5)³ = −125, so ³√−125 = −5
c) 0.1³ = 0.001, so ³√0.001 = 0.1

Finding cube roots without a calculator

For a bigger number, use prime factorisation and split the factors into three identical groups. One group is the cube root.

Example 12: ³√216 (textbook p. 64)

Factor tree: 216 = 12 × 18, 12 = 3 × 4, 4 = 2 × 2, 18 = 3 × 6, 6 = 3 × 2

216 = 2 × 2 × 2 × 3 × 3 × 3
= (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6
³√216 = 6

Example 13: fractions and decimals (textbook pp. 64–65)

³√8125 = ³√8³√125 = 25
³√−81192 = ³√−2764 = −34 (simplify first)
³√338 = ³√278 = 32 = 112 (improper fraction first)
³√0.027 = ³√0.3 × 0.3 × 0.3 = 0.3
³√−0.008 = ³√−81000 = −210 = −0.2

³√0.027 is 0.3, not 0.03 and not 0.009 (0.027 ÷ 3). A cube root is not "divide by 3". Check: 0.3 × 0.3 × 0.3 = 0.027 ✓

Cube roots with a calculator

Question	Press	Answer (2 d.p.)
³√24	∛ 2 4 =	2.88
³√−104.8	∛ (−) 1 0 4 . 8 =	−4.71
³√−129	∛ (−) 1 a b/c 2 a b/c 9 =	−1.07

Estimating cubes and cube roots

Example 14: estimate 4.2³ and ³√180 (textbook p. 66)

4.2 is between 4 and 5, so 4.2³ is between 4³ = 64 and 5³ = 125. 4.2 is nearer 4, so 4.2³ ≈ 64.
180 is between the perfect cubes 125 and 216, so ³√180 is between 5 and 6. 180 is nearer 216, so ³√180 ≈ 6.

Number line: 125 at 5, 216 at 6, 180 at the cube root of 180, closer to 6 — ³√180 lies between 5 and 6, closer to 6

a) Find ³√15 correct to 2 decimal places. b) Estimate ³√69.

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a) ³√15 = 2.466… = 2.47 (2 d.p.)
b) 69 is between 64 and 125, much nearer 64, so ³√69 ≈ 4

Solving problems

Example 15: the sculptor (textbook pp. 66–67)

A sculptor carves a small cube out of a wooden cube with sides of 6 cm. The remaining block has a volume of 189 cm³. Find the side of the small cube.

Wooden cube of side 6 cm with a small cube cut out of the top corner; remaining block 189 cm³

Volume of the wooden cube = 6³ = 216 cm³
Volume of the small cube = 216 − 189 = 27 cm³
Side of the small cube = ³√27 = 3 cm

Check: 3³ + 189 = 27 + 189 = 216 = 6³ ✓

Malik has 150 cm of wire. He wants to make the frame of a cube with a volume of 2 197 cm³. Is the wire long enough?

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Side = ³√2 197 = 13 cm. A cube has 12 edges: 12 × 13 = 156 cm. 156 cm > 150 cm, so the wire is not long enough.

Combined operations

Work in this order:

Find the squares, square roots, cubes and cube roots.
Work out the brackets.
× and ÷ from left to right.
+ and − from left to right.

Example 16 (textbook p. 68)

0.5² + ³√1 000 = 0.25 + 10 = 10.25
(−3)³ − √64 = −27 − 8 = −35
√25 + (−0.2)² ÷ ³√0.008 = 5 + 0.04 ÷ 0.2 = 5 + 0.2 = 5.2 (÷ before +)
³√−278 × (√36 − 2³)² = −32 × (6 − 8)² = −32 × 4 = −6

In 5 + 0.04 ÷ 0.2, do the division first. (5 + 0.04) ÷ 0.2 = 25.2 is wrong.

a) √49 + 3² b) ³√27 − 1.5²

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a) 7 + 9 = 16
b) 3 − 2.25 = 0.75

Summary

	Meaning	Example	Perfect ones
Square	a × a = a²	7² = 49	1, 4, 9, 16, 25, 36, …
Square root	√a × a = a	√49 = 7	Prime factors split into 2 identical groups
Cube	a × a × a = a³	5³ = 125	1, 8, 27, 64, 125, …
Cube root	³√a × a × a = a	³√125 = 5	Prime factors split into 3 identical groups

Remember	Key fact
Signs	(negative)² is positive; (negative)³ is negative; ³√negative is negative
Fractions	Simplify, or change a mixed number to an improper fraction, before finding the root
Decimals	0.3² = 0.09; 0.2³ = 0.008; √0.36 = 0.6; ³√0.027 = 0.3
Estimating	Find the perfect squares (or cubes) on either side and choose the nearer one
Multiplying roots	√a × √a = a; √a × √b = √ab
Order	Powers and roots, then brackets, then × ÷, then + −
Area and volume	Side of a square = √area; side of a cube = ³√volume