Square Roots · Evaluate & estimate · 2.4

Estimate a root.

To estimate an irrational root, trap it between the two nearest perfect squares. 4² = 16 and 5² = 25, and 16 < 20 < 25, so √20 is between 4 and 5 (and closer to 4½).

01 / THE IDEA

See it work.

Watch the method

Estimate a root

16 < 20 < 25, so √20 is between 4 and 5.

1 / 2
20\sqrt{20}
1

Trap it between squares

4² = 16 and 5² = 25, and 16 < 20 < 25.

42<20<524^2 < 20 < 5^2
2

Read off the estimate

So √20 is between 4 and 5.

4<20<54 < \sqrt{20} < 5
— / WHY IT WORKS

The idea underneath.

Between squares

Find the perfect squares just below and just above the radicand; the root lies between their roots.

Why it works

√20 sits between √16 = 4 and √25 = 5.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

√24 is between which two whole numbers?

1

Answer

4 and 5

EX 2medium

√15 is between which two whole numbers?

1

Answer

3 and 4

EX 3hard

√27 is between which two whole numbers?

1

Answer

5 and 6

04 / COMMON TRAPS

Where students slip.

Trap

Guessing without the bracketing squares.

Fix

Use the nearest perfect squares below and above.

Trap

Picking the wrong pair.

Fix

The radicand must fall between the two squares.

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

√20 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
C2

√12 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
C3

√6 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
C4

√20 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
Apply

Apply (4)

A1

√5 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
A2

√20 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
A3

√2 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
A4

√27 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
Extension

Extension (4)

E1

√2 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
E2

√10 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
E3

√7 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.
E4

√5 is between which two whole numbers?

Need a hint?
  • Find the perfect squares around the radicand.
  • The root is between their roots.