Maths · 1 Number

1.3 Powers and roots

Before you start

Times tables up to $12 \times 12$ , and multiplying a two-digit number by a one-digit number without a calculator.
Multiplying negative numbers: a negative times a negative is positive, e.g. $(- 4) \times (- 4) = 16$ .
Multiplying fractions (top times top, bottom times bottom) and turning a mixed number into an improper fraction, e.g. $1 \frac{1}{2} = \frac{3}{2}$ .
Writing a whole number as a product of its prime factors in index form, e.g. $72 = 2^{3} \times 3^{2}$ (subtopic 1.1).
Rounding to 3 significant figures.

Syllabus objectives covered (1)

Calculate with the following: • squares • square roots • cubes • cube roots • other powers and roots of numbers.

Part 1 of 7

Squares and square roots

To square a number, multiply it by itself. $7^{2}$ means $7 \times 7 = 49$ , and you read it as '7 squared'. The small raised 2 is the power, also called the index. It counts how many 7s are multiplied together. It does not mean 'times 2'.
A square root undoes squaring. $\sqrt{49}$ asks: which number, multiplied by itself, gives 49? The answer is $7$ , because $7^{2} = 49$ . You can always check a square root by squaring your answer.
The syllabus expects you to know the squares of 1 to 15, and their square roots, without a calculator. Learn the list in both directions, so that $13^{2}$ gives you 169 and $\sqrt{169}$ gives you 13.
$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$
If you forget one of the larger squares, rebuild it by splitting the multiplication into tens and units: $14 \times 14 = 14 \times 10 + 14 \times 4 = 140 + 56 = 196$ .
A negative number squared is positive: ${(- 6)}^{2} = (- 6) \times (- 6) = 36$ . So 36 has two square roots, $6$ and $- 6$ . The root sign on its own means the positive one only: $\sqrt{36} = 6$ . If a question asks for both square roots, write $\pm 6$ .
No number squared gives a negative answer, so $\sqrt{- 9}$ has no value. Do not write $- 3$ , because ${(- 3)}^{2} = + 9$ .
For a multiple of 10, square the front digits and double the number of zeros: $30^{2} = 900$ and $40^{2} = 1600$ . To go back, halve the number of zeros: $\sqrt{3600} = 60$ and $\sqrt{4900} = 70$ .
The trap: halving the zeros only works when the number of zeros is even. $\sqrt{490}$ is not 70, because $70^{2} = 4900$ . In fact $\sqrt{490}$ is about $22.1$ .

Part 2 of 7

Cubes and cube roots

To cube a number, use it three times in a multiplication. $4^{3} = 4 \times 4 \times 4$ . Do it in two steps: $4 \times 4 = 16$ , then $16 \times 4 = 64$ . The name comes from volume: a cube with edges of 4 cm has a volume of $64$ cm³.
The trap is to read the 3 as 'times 3'. $4^{3}$ is $64$ , not $12$ .
A cube root undoes cubing. $\sqrt[3]{64}$ asks: which number, used three times in a multiplication, gives 64? The answer is $4$ . The small 3 in the crook of the root sign tells you it is a cube root and not a square root.
The syllabus expects you to know these cubes and their cube roots without a calculator.
$1^{3} = 1, 2^{3} = 8, 3^{3} = 27, 4^{3} = 64, 5^{3} = 125, 10^{3} = 1000$
Here negative numbers behave differently from squares. ${(- 2)}^{3} = (- 2) \times (- 2) \times (- 2)$ . The first two give $+ 4$ , and then $4 \times (- 2) = - 8$ . A negative number cubed is negative.
So a negative number does have a cube root: $\sqrt[3]{- 8} = - 2$ and $\sqrt[3]{- 125} = - 5$ . Compare this with $\sqrt{- 9}$ , which has no value. A square can never be negative, but a cube can.
For a multiple of 10, cube the front digits and triple the number of zeros: $20^{3} = 8000$ . To go back, divide the number of zeros by 3: $\sqrt[3]{8000} = 20$ and $\sqrt[3]{1 000 000} = 100$ .

Part 3 of 7

Other powers and roots

The same notation works for any whole-number power. $a^{n}$ means $n$ copies of $a$ multiplied together. $a$ is the base and $n$ is the index.
$2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$
Build a power one step at a time, multiplying by the base each time, so that a slip shows up at once. Powers of 2: $2, 4, 8, 16, 32, 64$ . Powers of 3: $3, 9, 27, 81, 243$ .
Powers of 10 are the easy case: $10^{n}$ is 1 followed by $n$ zeros. $10^{6} = 1 000 000$ .
An nth root undoes the $n$ th power. $\sqrt[n]{x}$ is the number that gives $x$ when you raise it to the power $n$ . $\sqrt[4]{81} = 3$ because $3^{4} = 81$ , and $\sqrt[5]{32} = 2$ because $2^{5} = 32$ .
To find one without a calculator, work up the powers of 2, 3, 4, 5, … until you reach the number. For $\sqrt[4]{625}$ : $2^{4} = 16$ , $3^{4} = 81$ , $4^{4} = 256$ , $5^{4} = 625$ . So $\sqrt[4]{625} = 5$ .
Signs follow a pattern. A negative base with an even power gives a positive answer: ${(- 2)}^{4} = 16$ . With an odd power it gives a negative answer: ${(- 3)}^{3} = - 27$ and ${(- 1)}^{7} = - 1$ . Count the negative signs: each pair of them makes a positive.
So an odd root of a negative number exists, e.g. $\sqrt[5]{- 32} = - 2$ . An even root of a negative number, such as $\sqrt[4]{- 16}$ , has no value. Like the square root sign, an even root sign means the positive root: $\sqrt[4]{16} = 2$ .
Zero, negative and fractional indices, such as $7^{- 2}$ or $81^{\frac{1}{2}}$ , belong to the indices subtopic (1.7). This lesson uses whole-number powers only.

Part 4 of 7

Powers and roots of fractions and decimals

To square a fraction, square the top and the bottom. This is just fraction multiplication: $\frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3}$ . Cubing works the same way.
${(\frac{2}{3})}^{2} = \frac{4}{9} {(\frac{2}{5})}^{3} = \frac{8}{125}$
A root of a fraction works the same way: take the root of the top and the root of the bottom. $\sqrt{\frac{25}{36}} = \frac{5}{6}$ and $\sqrt[3]{\frac{8}{27}} = \frac{2}{3}$ . Check: $\frac{5}{6} \times \frac{5}{6} = \frac{25}{36}$ .
A mixed number must become an improper fraction first. $1 \frac{1}{2} = \frac{3}{2}$ , so ${(1 \frac{1}{2})}^{2} = \frac{9}{4} = 2 \frac{1}{4}$ . In the same way, $2 \frac{7}{9} = \frac{25}{9}$ , so $\sqrt{2 \frac{7}{9}} = \frac{5}{3} = 1 \frac{2}{3}$ .
For a decimal, square the digits and then count the decimal places. Squaring doubles the number of places and cubing triples it. ${0.3}^{2} = 0.09$ , with 2 places. ${1.2}^{2} = 1.44$ . ${0.2}^{3} = 0.008$ , with 3 places.
The places double because $0.3 = \frac{3}{10}$ , so ${0.3}^{2} = \frac{9}{100}$ , which is $0.09$ .
Roots reverse this. A square root halves the number of decimal places: $\sqrt{0.09} = 0.3$ and $\sqrt{0.0016} = 0.04$ . A cube root divides it by 3: $\sqrt[3]{0.008} = 0.2$ and $\sqrt[3]{0.001} = 0.1$ .
The trap: $\sqrt{0.9}$ is not $0.3$ . The number 0.9 has one decimal place, and one does not halve. Check: ${0.3}^{2} = 0.09$ , not $0.9$ . A calculator gives $\sqrt{0.9} = 0.9486 \dots$
A sense check for numbers between 0 and 1: squaring makes them smaller, and taking the square root makes them bigger. ${0.5}^{2} = 0.25$ , which is less than 0.5. If the square of a decimal below 1 comes out bigger, a decimal place has gone missing.

Part 5 of 7

Powers and roots inside a longer calculation

Powers and roots come before multiplying, dividing, adding and subtracting. Only brackets come earlier. So work out each power and each root first, then combine the values.
$4^{2} \times \sqrt[3]{27} = 16 \times 3 = 48$
A power belongs only to the number it touches. $2 \times 3^{2} = 2 \times 9 = 18$ . It is not $6^{2} = 36$ . If the whole product were squared, it would be written ${(2 \times 3)}^{2}$ .
The same goes for a minus sign. $- 3^{2}$ means 'the negative of $3^{2}$ ', which is $- 9$ . ${(- 3)}^{2}$ means $(- 3) \times (- 3)$ , which is $9$ . When you square a negative number on a calculator, type the brackets as well.
The bar of a root sign acts as a bracket. Work out everything under it first. $\sqrt{9 + 16} = \sqrt{25} = 5$ .
$\sqrt{9 + 16} = 5 but \sqrt{9} + \sqrt{16} = 3 + 4 = 7$
A power does not share out over a sum either. ${(3 + 4)}^{2} = 7^{2} = 49$ , but $3^{2} + 4^{2} = 9 + 16 = 25$ . Add inside the bracket first.
On a calculator, use the square key, the root keys and the power key (marked ^ or $x^{y}$ , depending on the model). Put brackets around everything under a root sign, and around the whole top and the whole bottom of a fraction.
Write down the full calculator display before you round. Then round only the final answer, to 3 significant figures unless the question says otherwise. If you round halfway through, the final answer can come out wrong.

Part 6 of 7

Roots of larger numbers from prime factors

To find the square root of a large number without a calculator, write the number as a product of primes first. For example, $3136 = 2^{6} \times 7^{2}$ .
A square root splits a number into two equal factors. Six 2s give three 2s to each factor, and two 7s give one 7 to each factor.
$3136 = (2^{3} \times 7) \times (2^{3} \times 7) \Rightarrow \sqrt{3136} = 2^{3} \times 7 = 56$
So for a square root, halve every index. A cube root splits the number into three equal factors, so divide every index by 3. $2744 = 2^{3} \times 7^{3}$ , so $\sqrt[3]{2744} = 2 \times 7 = 14$ .
This also tells you whether a number is a square number or a cube number at all. It is a square only if every index is even. It is a cube only if every index is a multiple of 3. $72 = 2^{3} \times 3^{2}$ is neither: the index 3 is odd, and the index 2 is not a multiple of 3.
Always check by multiplying back: $56 \times 56 = 3136$ .

Part 7 of 7

Roots that are not whole numbers

Most numbers are not square numbers, so most square roots are not whole. You can still say which two whole numbers a root lies between, by using the squares you know. $49 < 50 < 64$ , so $\sqrt{50}$ lies between $7$ and $8$ .
It is much nearer 7, because 50 is only 1 more than 49. A calculator gives $\sqrt{50} = 7.071 \dots$ , which is $7.07$ to 3 significant figures.
Cube roots work the same way, with the cubes you know. $64 < 100 < 125$ , so $\sqrt[3]{100}$ lies between 4 and 5. A calculator gives $4.6415 \dots$ , which is $4.64$ to 3 significant figures.
Use this estimate to check every root you find on a calculator. If you get $\sqrt{50} = 25$ , you divided by 2 instead of taking the root. The estimate 'between 7 and 8' shows the error at once.
Roots appear in context when you work backwards from an area or a volume. A square of area 30 cm² has side $\sqrt{30} = 5.48$ cm (3 s.f.). A cube of volume 200 cm³ has edge $\sqrt[3]{200} = 5.85$ cm (3 s.f.).

Worked examples

Example 1. Without a calculator, work out (a) $\sqrt{196}$ (b) $4^{3} - \sqrt{81}$ (c) $3^{2} \times \sqrt[3]{1000}$ (d) ${(- 2)}^{5}$ .

Show the working

(a) 196 is in the list of squares: $14^{2} = 196$ . So $\sqrt{196} = 14$ . Check: $14 \times 14 = 140 + 56 = 196$ .
(b) Do the power and the root first, because both come before subtracting. $4^{3} = 64$ and $\sqrt{81} = 9$ .
$64 - 9 = 55$
(c) $3^{2} = 9$ , and $\sqrt[3]{1000} = 10$ because $10^{3} = 1000$ . Then multiply.
$9 \times 10 = 90$
(d) Build the power one step at a time: $- 2, 4, - 8, 16, - 32$ . The power 5 is odd, so the answer is negative.

Answer: (a) $14$ (b) $55$ (c) $90$ (d) $- 32$

Example 2. Without a calculator, work out (a) ${(\frac{3}{4})}^{2}$ (b) $\sqrt{1 \frac{9}{16}}$ (c) ${0.4}^{3}$ (d) $\sqrt{0.0049}$ .

Show the working

(a) Square the top and the bottom.
${(\frac{3}{4})}^{2} = \frac{3^{2}}{4^{2}} = \frac{9}{16}$
(b) Change the mixed number to an improper fraction first: $1 \frac{9}{16} = \frac{16 + 9}{16} = \frac{25}{16}$ . Then take the root of the top and of the bottom.
$\sqrt{\frac{25}{16}} = \frac{5}{4} = 1 \frac{1}{4}$
Check (b): $\frac{5}{4} \times \frac{5}{4} = \frac{25}{16}$ . Notice that the answer is not $1 \frac{3}{4}$ . Taking the root of the fraction part alone gives a wrong answer.
(c) $4^{3} = 64$ . The number $0.4$ has 1 decimal place, and cubing triples that to 3 places.
${0.4}^{3} = 0.064$
(d) $0.0049$ has 4 decimal places, so its square root has 2. The digits 49 have square root 7.
$\sqrt{0.0049} = 0.07$
Check (d): $0.07 \times 0.07 = 0.0049$ . Correct.

Answer: (a) $\frac{9}{16}$ (b) $\frac{5}{4} = 1 \frac{1}{4}$ (c) $0.064$ (d) $0.07$

Example 3. You are given that $3136 = 2^{6} \times 7^{2}$ and $2744 = 2^{3} \times 7^{3}$ . Without a calculator, find (a) $\sqrt{3136}$ (b) $\sqrt[3]{2744}$ .

Show the working

(a) A square root splits the number into two equal factors. Share the primes equally between two groups: the six 2s give three to each group, and the two 7s give one to each group.
$3136 = (2^{3} \times 7) \times (2^{3} \times 7)$
The square root is one of those groups.
$\sqrt{3136} = 2^{3} \times 7 = 8 \times 7 = 56$
(b) A cube root splits the number into three equal factors. The three 2s give one to each group, and the three 7s give one to each group.
$\sqrt[3]{2744} = 2 \times 7 = 14$
Check both by multiplying back: $56 \times 56 = 3136$ , and $14 \times 14 \times 14 = 196 \times 14 = 2744$ .

Answer: (a) $56$ (b) $14$

Example 4. A square garden has an area of 40 m². (a) Without a calculator, state the two whole numbers that the side length of the garden lies between. (b) Calculate the side length, giving your answer correct to 3 significant figures. (c) Calculate $\frac{{2.7}^{3} + \sqrt{19.5}}{{4.1}^{2}}$ , giving your answer correct to 3 significant figures.

Show the working

(a) Side × side = area, so the side length is $\sqrt{40}$ . The squares on either side of 40 are $36 = 6^{2}$ and $49 = 7^{2}$ . So $\sqrt{40}$ lies between 6 and 7.
(b) On the calculator, $\sqrt{40} = 6.324555 \dots$ Write this down, then round. The fourth significant figure is 4, so round down to $6.32$ . This is between 6 and 7, so it agrees with (a).
(c) Type the whole top in brackets and the whole bottom in brackets: (2.7^3 + √(19.5)) ÷ (4.1^2). Without the brackets, the calculator divides only the $\sqrt{19.5}$ by ${4.1}^{2}$ .
If you prefer to work in stages, find each piece and keep every digit. Top: $19.683 + 4.415880 \dots = 24.098880 \dots$ Bottom: ${4.1}^{2} = 16.81$ .
$\frac{24.098880 \dots}{16.81} = 1.433603 \dots$
Round only now. The fourth significant figure is 3, so round down to $1.43$ .

Answer: (a) Between $6$ and $7$ (b) $6.32$ m (c) $1.43$

Where people go wrong

Mistake: Multiplying by the power, or halving for a square root: writing $5^{3} = 15$ or $\sqrt{16} = 8$ .

Instead: $5^{3} = 5 \times 5 \times 5 = 125$ . $\sqrt{16} = 4$ , because $4^{2} = 16$ .

The power counts how many copies of the number are multiplied together. It is not a number to multiply by. A root asks which number, multiplied by itself, gives the value. Squaring back catches the error: $8^{2} = 64$ , not 16.

Mistake: Taking the root of each part of a sum: $\sqrt{9 + 16} = 3 + 4 = 7$ .

Instead: Add first: $\sqrt{9 + 16} = \sqrt{25} = 5$ .

The bar of the root sign is a bracket, so its contents are worked out first. A root does not share out over $+$ or $-$ . Check: $7^{2} = 49$ , not 25.

Mistake: Losing a decimal place: ${0.3}^{2} = 0.9$ , or $\sqrt{0.9} = 0.3$ .

Instead: ${0.3}^{2} = 0.09$ . Squaring doubles the number of decimal places. $\sqrt{0.09} = 0.3$ , but $\sqrt{0.9} = 0.9486 \dots$

$0.3 \times 0.3 = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100}$ . The answer is hundredths, not tenths.

Mistake: Giving a positive answer for a negative number cubed, or saying that $\sqrt[3]{- 27}$ has no value.

Instead: ${(- 3)}^{3} = - 27$ , so $\sqrt[3]{- 27} = - 3$ . Only even roots of negative numbers, such as $\sqrt{- 9}$ , have no value.

The first two negatives multiply to a positive, and the third negative makes the answer negative again. A square is never negative, but a cube can be.

Mistake: Squaring a mixed number one part at a time: ${(1 \frac{1}{2})}^{2} = 1 \frac{1}{4}$ .

Instead: Convert first: $1 \frac{1}{2} = \frac{3}{2}$ , so ${(\frac{3}{2})}^{2} = \frac{9}{4} = 2 \frac{1}{4}$ .

$1 \frac{1}{2}$ means $1 + \frac{1}{2}$ , and a power does not share out over a sum. A sense check: ${1.5}^{2}$ must be bigger than $1.5$ , so $1 \frac{1}{4}$ cannot be right.

Check yourself

Without a calculator, write down $\sqrt{225}$ and $13^{2}$ .
Answer
$\sqrt{225} = 15$ and $13^{2} = 169$ .
Without a calculator, write down $\sqrt[3]{64}$ and $\sqrt[3]{1000}$ .
Answer
$\sqrt[3]{64} = 4$ and $\sqrt[3]{1000} = 10$ .
Work out $2^{6}$ and $\sqrt[4]{81}$ .
Answer
$2^{6} = 64$ (build it: $2, 4, 8, 16, 32, 64$ ). $\sqrt[4]{81} = 3$ , because $3^{4} = 81$ .
Work out ${(- 4)}^{3}$ . Does $\sqrt{- 16}$ have a value?
Answer
${(- 4)}^{3} = - 64$ . $\sqrt{- 16}$ has no value, because no number squared gives a negative answer.
Work out ${(\frac{2}{5})}^{2}$ and $\sqrt{\frac{36}{49}}$ .
Answer
${(\frac{2}{5})}^{2} = \frac{4}{25}$ and $\sqrt{\frac{36}{49}} = \frac{6}{7}$ .
Without a calculator, work out ${0.6}^{2}$ and $\sqrt{0.0081}$ .
Answer
${0.6}^{2} = 0.36$ (1 place doubles to 2). $\sqrt{0.0081} = 0.09$ (4 places halve to 2; check ${0.09}^{2} = 0.0081$ ).
Work out $\sqrt{36 + 64}$ and $\sqrt{36} + \sqrt{64}$ . Are they equal?
Answer
$\sqrt{36 + 64} = \sqrt{100} = 10$ , but $\sqrt{36} + \sqrt{64} = 6 + 8 = 14$ . They are not equal.
Between which two whole numbers does $\sqrt[3]{50}$ lie? Then use a calculator to find it correct to 3 significant figures.
Answer
Between 3 and 4, because $27 < 50 < 64$ . $\sqrt[3]{50} = 3.684 \dots = 3.68$ (3 s.f.).

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Write down the value of $\sqrt{144}$ . 1 mark No calculator

Mark scheme

B1 12 cao

(b) Write down the value of $\sqrt[3]{125}$ . 1 mark No calculator

Mark scheme

B1 5 cao

(c) Work out $2^{4} \times \sqrt{49}$ . 2 marks No calculator

Mark scheme

M1 16 and 7 both seen, or 16 × 7
A1 112 cao

(d) Work out ${(- 3)}^{3} + 4^{2}$ . 2 marks No calculator

Mark scheme

M1 −27 and 16 both seen, or −27 + 16
A1 −11 cao

Question 2.

(a) Work out ${0.3}^{3}$ . 1 mark No calculator

Mark scheme

B1 0.027 oe, e.g. 27/1000

(b) Write $\sqrt{\frac{49}{64}}$ as a fraction. 1 mark No calculator

Mark scheme

B1 7/8 cao

(c) Find the value of $\sqrt{0.36} + \sqrt[3]{0.008}$ . 2 marks No calculator

Mark scheme

M1 0.6 or 0.2 seen, oe e.g. 6/10 or 2/10
A1 0.8 oe, e.g. 4/5

Question 3. You are given that $5184 = 2^{6} \times 3^{4}$ .

(a) Without using a calculator, find $\sqrt{5184}$ . 2 marks No calculator

Mark scheme

M1 2³ × 3² oe seen, e.g. 8 × 9, or 5184 = (2³ × 3²) × (2³ × 3²)
A1 72 cao

(b) Explain why $5184$ is not a cube number. 1 mark No calculator

Mark scheme

B1 the index of 3 is 4, which is not a multiple of 3, so 5184 cannot be split into three equal factors oe; or 17³ = 4913 and 18³ = 5832, so 5184 lies between two consecutive cubes

Question 4. A cube has a volume of 250 cm³.

(a) Calculate the length of one edge of the cube. Give your answer correct to 3 significant figures. 2 marks

Mark scheme

M1 ∛250 seen
A1 6.30 or more accurate value 6.2996… (6.299 to 6.300)

(b) Calculate the area of one face of the cube. 2 marks

Mark scheme

M1 (their edge)² seen, e.g. 6.2996…² or 6.30², or 250 ÷ (their edge), or 250^(2/3); FT their (a)
A1 awrt 39.7

(c) Calculate $\sqrt[4]{250} + {1.5}^{5}$ . Give your answer correct to 3 significant figures. 2 marks

Mark scheme

M1 3.976… or 7.59375 seen
A1 11.6 (11.570…)

How this is examined

Finding the side of a square or the edge of a cube from its area or volume, and then using it: A calculator question with a short context, such as a square field or a cube-shaped box, in two linked parts: a root first, then a calculation that uses the root The root of the given value written down ( $\sqrt[3]{250}$ , say) for the method mark, then the value to 3 s.f. with its unit. In the next part, use the unrounded value from the calculator, so the final answer rounds correctly.
Working out an expression that mixes recalled squares, cubes and roots, without a calculator: A 2-mark Work out on the non-calculator paper, such as a power times a cube root, or a cube minus a square root Each power and root replaced by its value (e.g. $16$ and $7$ for $2^{4} \times \sqrt{49}$ ) for the method mark. Then the single correct value, here $112$ . Do the powers and roots before you multiply, add or subtract.
Finding a square root or cube root of a large number from its prime factors: A 2-mark Find on the non-calculator paper, with the prime factorisation given in the question The root written as a product with every index halved (or divided by 3 for a cube root), e.g. $2^{3} \times 3^{2}$ , then that product worked out as one number, $72$ . The product shows the method; the single number is the answer.
Calculating a long expression with powers and roots on the calculator: A 2-mark Calculate with a fraction line, a root and a power, and the answer to 3 significant figures Brackets around the whole top and the whole bottom, the full display written down (e.g. $1.433603 \dots$ ), then the answer to 3 s.f., $1.43$ . The written display shows the calculation was right, even if the rounding then goes wrong.
Powers and roots of fractions and decimals, without a calculator: A 1- or 2-mark Work out with a decimal such as ${0.3}^{3}$ or $\sqrt{0.36}$ , or a Write with a fraction such as $\sqrt{\frac{49}{64}}$ The decimal point placed by counting places: ${0.3}^{3} = 0.027$ (three places), $\sqrt{0.36} = 0.6$ (half of two places). For a fraction, the root of the top over the root of the bottom: $\frac{7}{8}$ .
Writing down a square, cube or root that you must know by heart: A 1-mark Write down, such as the value of $\sqrt{196}$ or $\sqrt[3]{125}$ , done without a calculator The exact whole number and nothing else; no working is needed. The syllabus requires recall of the squares of 1 to 15 and the cubes of 1, 2, 3, 4, 5 and 10, with their roots.

Words to know

Square: A number multiplied by itself, e.g. $7^{2} = 7 \times 7 = 49$ .
Square root: The number that, multiplied by itself, gives the value. $\sqrt{49} = 7$ . The sign $\sqrt{}$ means the positive root.
Cube: A number used three times in a multiplication, e.g. $4^{3} = 4 \times 4 \times 4 = 64$ .
Cube root: The number that, cubed, gives the value. $\sqrt[3]{64} = 4$ and $\sqrt[3]{- 8} = - 2$ .
Power (index): The small raised number that counts how many copies of the base are multiplied together. In $2^{5}$ the index is 5.
Base: The number being multiplied by itself in a power. In $2^{5}$ the base is 2.
nth root: The number that, raised to the power $n$ , gives the value. $\sqrt[4]{81} = 3$ because $3^{4} = 81$ .
Square number: A whole number that is the square of a whole number: $1, 4, 9, 16, \dots$
Cube number: A whole number that is the cube of a whole number: $1, 8, 27, 64, \dots$

All Maths lessons Next: 1.4 Fractions, decimals and percentages