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Ignition Learning — Activity Sheet

Surds

Mathematics · Year 10

Name: ______________________Date: ____________

A surd is a square root (or other root) that doesn't simplify to a whole number, like √2 or √7 — it stays in exact, irrational form rather than being rounded to a decimal. Surds can often be simplified by finding a square factor: √12 = √(4×3) = √4 × √3 = 2√3. Like surds (with the same number under the root) can be added or subtracted, similar to like terms in algebra.

Example

√12 simplifies to 2√3, since 12 = 4 × 3 and √4 = 2. Then 2√3 + 3√3 = 5√3, combining like surds — but 2√3 + 3√5 cannot be combined further, since they have different numbers under the root.

Key terms

Surd:
A root (like a square root) that doesn't simplify to a whole number.
Simplify:
Rewriting a surd using its largest square factor.
Like surds:
Surds with the same number under the root, which can be combined.

Questions

  1. 1. A surd is:

    • A root that doesn't simplify to a whole number
    • Always a whole number
    • A type of fraction only
    • A type of percentage

  2. 2. Which of these is a surd?

    • √2
    • √4
    • √9
    • √16

  3. 3. Which of these is NOT a surd (it simplifies to a whole number)?

    • √25
    • √3
    • √7
    • √11

  4. 4. √4 equals:

    • 2
    • 4
    • 16
    • 1

  5. 5. √9 equals:

    • 3
    • 9
    • 81
    • 1

  6. 6. Like surds can be:

    • Added or subtracted
    • Never combined in any way
    • Only multiplied, never added
    • Ignored completely

  7. 7. 2√3 + 3√3 equals:

    • 5√3
    • 6√3
    • 5√6
    • 6√6

  8. 8. Simplify √12

    • 2√3
    • 4√3
    • 2√6
    • 3√4

  9. 9. Simplify √8

    • 2√2
    • 4√2
    • 2√4
    • 8√1

  10. 10. Simplify √18

    • 3√2
    • 2√3
    • 6√3
    • 9√2

  11. 11. Simplify √50

    • 5√2
    • 2√5
    • 10√5
    • 25√2

  12. 12. 5√2 − 2√2 equals:

    • 3√2
    • 3√4
    • 7√2
    • 7√4

  13. 13. Simplify √27

    • 3√3
    • 9√3
    • 3√9
    • 27√1

  14. 14. Can 2√3 and 3√5 be combined by adding them together?

    • No, since they have different numbers under the root
    • Yes, they combine to 5√8
    • Yes, they combine to 5√15
    • Yes, they combine to 6√3

  15. 15. Simplify √72

    • 6√2
    • 2√6
    • 3√8
    • 8√3

  16. 16. Simplify 3√8 + 2√2

    • 8√2
    • 5√10
    • 5√2
    • 6√2

  17. 17. A square has an area of 32cm². What is its exact side length, simplified?

    • 4√2 cm
    • 2√8 cm
    • 16√2 cm
    • 8√4 cm

  18. 18. Simplify √45 − √20

    • √5
    • 3√5
    • 2√5
    • 25

  19. 19. Why do exact surd answers (like 2√3) sometimes get used instead of decimal approximations in maths?

    • Surds preserve exact precision, avoiding rounding errors in further calculations
    • Decimals are always more accurate than surds
    • Surds are always easier to calculate mentally
    • Exact and decimal forms are always identical

  20. 20. Simplify √98 + √8

    • 9√2
    • 7√2 + 2√2
    • 10√2
    • Both 9√2 and 7√2+2√2 are correct

  21. 21. Simplify √75

    • 5√3
    • 3√5
    • 15√5
    • 25√3

Answer key (parent copy)

  1. 1. A root that doesn't simplify to a whole number
  2. 2. √2
  3. 3. √25
  4. 4. 2
  5. 5. 3
  6. 6. Added or subtracted
  7. 7. 5√3
  8. 8. 2√3
  9. 9. 2√2
  10. 10. 3√2
  11. 11. 5√2
  12. 12. 3√2
  13. 13. 3√3
  14. 14. No, since they have different numbers under the root
  15. 15. 6√2
  16. 16. 8√2
  17. 17. 4√2 cm
  18. 18. √5
  19. 19. Surds preserve exact precision, avoiding rounding errors in further calculations
  20. 20. Both 9√2 and 7√2+2√2 are correct
  21. 21. 5√3