Square Roots · Meet square roots · 1.3

A root is a side length.

A square root is literally the side of a square. A square with area 36 has side √36 = 6, because 6 × 6 = 36. Area → side is exactly the square root.

01 / THE IDEA

See it work.

The area model

A square root is a side length.

A square with area 36 has side √36 = 6.

side = √36 = 6
area = 36
= side × side
= 6 × 6

Watch the method

A root is a side length

A square of area 36 has side √36 = 6.

1 / 2
36\sqrt{36}
1

Ask the question

√36 asks: what number squared gives 36?

36=?\sqrt{36} = ?
2

Find the root

6² = 36, so √36 = 6.

36=6\sqrt{36} = 6
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Side from area

If a square’s area is A, its side is √A. This is why the operation is called a “square” root.

Why it works

Area 36 → 6 by 6 square.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Evaluate √64

1

Ask the question

√64 asks: what number squared gives 64? √64 = ?

2

Find the root

8² = 64, so √64 = 8. √64 = 8

EX 2medium

Evaluate √49

1

Ask the question

√49 asks: what number squared gives 49? √49 = ?

2

Find the root

7² = 49, so √49 = 7. √49 = 7

EX 3hard

Evaluate √144

1

Ask the question

√144 asks: what number squared gives 144? √144 = ?

2

Find the root

12² = 144, so √144 = 12. √144 = 12

04 / COMMON TRAPS

Where students slip.

Trap

Confusing area with perimeter.

Fix

The square root of the area gives the side, not the perimeter.

Trap

Dividing the area by 4.

Fix

The side is √area, not area ÷ 4.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Evaluate √4

Need a hint?
  • The side is √area.
  • Check: side × side = area.
C2

Evaluate √9

Need a hint?
  • The side is √area.
  • Check: side × side = area.
C3

Evaluate √81

Need a hint?
  • The side is √area.
  • Check: side × side = area.
C4

Evaluate √81

Need a hint?
  • The side is √area.
  • Check: side × side = area.
Apply

Apply (4)

A1

Evaluate √64

Need a hint?
  • The side is √area.
  • Check: side × side = area.
A2

Evaluate √100

Need a hint?
  • The side is √area.
  • Check: side × side = area.
A3

Evaluate √121

Need a hint?
  • The side is √area.
  • Check: side × side = area.
A4

Evaluate √121

Need a hint?
  • The side is √area.
  • Check: side × side = area.
Extension

Extension (4)

E1

Evaluate √144

Need a hint?
  • The side is √area.
  • Check: side × side = area.
E2

Evaluate √64

Need a hint?
  • The side is √area.
  • Check: side × side = area.
E3

Evaluate √4

Need a hint?
  • The side is √area.
  • Check: side × side = area.
E4

Evaluate √121

Need a hint?
  • The side is √area.
  • Check: side × side = area.