Square Roots · Evaluate & estimate · 2.3

Not every root is whole.

Most numbers are not perfect squares, so their roots are not whole numbers — they are irrational (endless, non-repeating decimals). √2 and √20 can’t be written exactly as fractions or terminating decimals.

01 / THE IDEA

See it work.

Watch the method

Not every root is whole

√20 is not a whole number — 20 isn’t a perfect square.

1 / 3
20\sqrt{20}
1

Split off the perfect square

20 = 4 × 5, and 4 is a perfect square.

20=4×5\sqrt{20} = \sqrt{4 \times 5}
2

Separate the roots

The square root of a product is the product of the roots.

=4⋅5= \sqrt{4} \cdot \sqrt{5}
3

Take the perfect-square root out

√4 = 2, which comes out front.

=25= 2\sqrt{5}
— / WHY IT WORKS

The idea underneath.

Irrational roots

If the radicand isn’t a perfect square, its square root is irrational: it can be estimated or left in radical form, but not written exactly as a decimal.

Why it works

√2 ≈ 1.414… never ends.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Is √20 a whole number (a perfect square)?

1

Answer

No

EX 2medium

Is √5 a whole number (a perfect square)?

1

Answer

No

EX 3hard

Is √64 a whole number (a perfect square)?

1

Answer

Yes

04 / COMMON TRAPS

Where students slip.

Trap

Rounding and calling it exact.

Fix

√2 ≈ 1.41, but it is not equal to 1.41.

Trap

Thinking every root is irrational.

Fix

Perfect squares have exact whole roots.

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

Is √121 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
C2

Is √81 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
C3

Is √81 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
C4

Is √20 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
Apply

Apply (4)

A1

Is √10 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
A2

Is √50 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
A3

Is √50 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
A4

Is √2 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
Extension

Extension (4)

E1

Is √8 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
E2

Is √100 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
E3

Is √49 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.
E4

Is √5 a whole number (a perfect square)?

Need a hint?
  • Is the radicand a perfect square?
  • If not, the root is irrational.