Square roots and cube roots

About this lesson

Grade 8 › Pythagorean theorem and irrational numbers

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Square roots and cube roots

Warm-up

Three quick questions from earlier lessons.

Discover

A square tile has an area of 49 cm². How long is a side?

  1. 7 cm
  2. 24.5 cm

Key idea

Square roots undo squaring; cube roots undo cubing.

a\sqrt{a} is the positive number whose square is aa (the side of a square of area aa). a3\sqrt[3]{a} is the number whose cube is aa (the edge of a cube of volume aa). If aa is not a perfect square, a\sqrt{a} lies between two consecutive whole numbers.

Worked example

30\sqrt{30} and 643\sqrt[3]{64}.

5<30<65 < \sqrt{30} < 6
25 < 30 < 36.
643=4\sqrt[3]{64} = 4
4 · 4 · 4 = 64.

Your turn, guided

Your turn: the edge of a cube of volume 125.

55
5³ = 125.

Practice ★

  1. A square has an area of 225 square units. How long is its side?
  2. A square has an area of 144 square units. How long is its side?
  1. A cube has a volume of 343 cubic units. How long is each edge?

Watch out!

Practice ★★

  1. 37\sqrt{ 37 } lies between two consecutive whole numbers. What is the smaller one?
  2. 17\sqrt{ 17 } lies between two consecutive whole numbers. What is the smaller one?

Exit ticket

Three questions to check you have got it.

  1. A square has an area of 225 square units. How long is its side?
  2. A square has an area of 144 square units. How long is its side?
  3. A square has an area of 49 square units. How long is its side?

Summary card

Square and cube roots

a\sqrt{a}: side of a square of area aa; a3\sqrt[3]{a}: edge of a cube of volume aa.

Watch out: a root is not half or a third.

End of the lesson

Next lesson: Rational and irrational numbers; repeating decimals Back to the chapter

Answers: Square roots and cube roots

A

  1. 1515
  2. 1212

B

  1. 77

C

  1. 66
  2. 44

D

  1. 1515
  2. 1212
  3. 77

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