Square roots and cube roots · lesson

Square roots

💡 The square root of a number is the positive number that times itself gives that number.

11 × 11 = 121, so √121 = 11. The square root sign √ asks: what number times itself makes this?

Perfect squares like 1, 4, 9, 16, 25, 36, 49 have whole-number roots. It helps to know them by heart up to 15 × 15 = 225.

For a fraction, take the root of the top and the bottom.

🧠 Let's work it out

Find √(4/25).

What do you think comes next?

💡 The cube root of a number is what you multiply by itself three times to get that number, and it can be negative.

6 × 6 × 6 = 216, so ∛216 = 6.

A negative times a negative times a negative is negative. So cube roots of negative numbers are negative: ∛(−8) = −2.

🧠 Let's work it out

Find ∛(−8).

What do you think comes next?

cube root: a number used three times as a factor to make the number you start with

💡 A square's side is the square root of its area, and a cube's edge is the cube root of its volume.

A square's area is side × side. So the side is the square root of the area.

A cube's volume is edge × edge × edge. So the edge is the cube root of the volume.

Once you have the side, you can find the perimeter or the area of a face.

🧠 Let's work it out

A square rug has an area of 196 square feet. How far is it around?

Side = √196 = 14 feet.
Perimeter = 4 × 14 = 56 feet.
Another example: a cube with volume 343 has edge ∛343 = 7, and a face has area 7 × 7 = 49.

What do you think comes next?

💡 Trap a square root between the perfect squares just below and just above the number.

Most square roots are not whole numbers. You can still find which whole numbers they sit between.

Find the perfect squares on each side. Their roots are your two whole numbers.

The root is closer to the one whose square is closer to your number.

🧠 Let's work it out

Estimate √20.

What do you think comes next?