Numbers and operations · Ages 13–14

Surds & irrational numbers worksheet

Covers: Square, cube and nth roots, Rational and irrational numbers, Simplifying surds, Working with surds, Constructing lengths √n

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Numbers and operations · Ages 13–14

Numsprite
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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.324 = 2^□ × 3^□, so 324 = 2^□ × 3^□ =
Fill in each blank
Answer:
2.Fill in two whole numbers in a row: < 21 <
Fill in each blank
Answer:
3.72 ÷ 2 =
Answer:
4.In simplest form: 3216 =
Fill in each blank
Answer:
5.A square-root spiral starts with shorter sides 1 and 1; each new right-angled triangle takes the last one's longest side and a side of 1 as its shorter sides. The 6th triangle's longest side is .
Answer:
6.A right-angled triangle with shorter sides 4 and has a longest side of 52.
Answer:

13.A cube has a volume of 1,728 cm³. What is its surface area, in cm²?
- A. 144
- B. 72
- C. 576
- D. 864
Choose the right answer
Working:
Answer:
14.Someone works out 144 and gets 72. What is wrong? What is the right answer?
1. (1)A square root means dividing by 2
2. (2)144 ÷ 2 = 72
Find the wrong step, then write the right answer
Working:
Answer:
15.Which of these four numbers is irrational?
- A. 0.7̇
- B. 25
- C. 0.1010010001…
- D. 100
Choose the right answer
Working:
Answer:
16.A square has an area of 50 cm². In simplest surd form, each side is cm.
Fill in each blank
Working:
Answer:
17.Rationalise the denominator of 205. Put the steps in order.
- A. 205 = (20 × 5) ÷ (5 × 5)
- B. 205 ÷ 5 = 45
- C. (20 × 5) ÷ (5 × 5) = 205 ÷ 5
Write the letters in the right order
Working:
Answer:
18.To find 41 on a number line, draw a right-angled triangle with shorter sides 5 and , then an arc about 0 with the longest side as radius.
Working:
Answer:

Show the answers and working

Surds & irrational numbers worksheet: Answers and working

1. 2, 4, 1, 2, 18
In primes, 324 = 2² × 3⁴; for the square root halve each index: 2¹ × 3² = 18.
2. 4, 5
Find the perfect squares either side of 21: 4² < 21 < 5².
3. 6
72 ÷ 2 = 72 ÷ 2 = 36 = 6.
4. 18, 6
216 = 66, so 3216 = 3 × 66 = 186.
5. 7
The 1st has 2, and each new triangle adds 1 under the root: the 6th has 7.
6. 6
4² + □² = 52, so □² = 52 − 4² and □ = 6.

7. True
64 has two square roots, 8 and −8, but the sign 64 means the positive one only, 8.
8. D. 3 and −3
3² = 9 and (−3)² = 9: both are square roots.
9. C. 94
94 isn't a perfect square, so 94's decimal never ends or repeats: irrational.
10. B. 28
5² < 28 < 6², so 5 < 28 < 6.
11. False
Combining like surds adds the numbers outside; the 6 inside stays.
12. C. 72
18 = 32 and 32 = 42; added, 72.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. D. 864
Edge = 31728 = 12 cm, surface area = 6 × 12² = 864 cm².
14. 12
A square root is the number whose square is 144: 12² = 144, so 144 = 12, not half of 144.
15. C. 0.1010010001…
0.1010010001… has more and more 0s between its 1s: it never ends or repeats, so it is irrational.
16. 5, 2
Side = 50 = 5² × 2 = 52 cm.
17. A → C → B
Multiply top and bottom by 5, making the bottom 5; then cancel, 20 ÷ 5 = 4.
18. 4
5² + □² = 41, so □² = 41 − 5² and □ = 4.

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