Square Roots · Simplify radicals · 3.5

Already simplest.

Some radicals can’t be simplified: if the radicand has no perfect-square factor bigger than 1, it’s already in simplest form. √15 = √15 (15 = 3 × 5, no square factor).

01 / THE IDEA

See it work.

Watch the method

Already simplest

√15 has no perfect-square factor, so it stays √15.

1 / 1
15\sqrt{15}
1

Already simplest

15 has no perfect-square factor bigger than 1, so √15 can't be simplified.

15\sqrt{15}
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number or a radical like 2√3 (you can write 2sqrt3).

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… =
— / WHY IT WORKS

The idea underneath.

Square-free radicands

When the radicand’s only perfect-square factor is 1, the radical is already simplest — leave it.

Why it works

15 = 3·5, neither repeated, so √15 stays.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify √5

1

Already simplest

5 has no perfect-square factor bigger than 1, so √5 can't be simplified. √5

EX 2medium

Simplify √2

1

Already simplest

2 has no perfect-square factor bigger than 1, so √2 can't be simplified. √2

EX 3hard

Simplify √27

1

Split off the perfect square

27 = 9 × 3, and 9 is a perfect square. √27 = \sqrt9 × 3

2

Separate the roots

The square root of a product is the product of the roots. = √9 · √3

3

Take the perfect-square root out

√9 = 3, which comes out front. = 3√3

04 / COMMON TRAPS

Where students slip.

Trap

Forcing a simplification that isn’t there.

Fix

If no square divides it, leave it under the √.

Trap

Trying to “simplify” √15 to a decimal.

Fix

Simplest radical form keeps it exact as √15.

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

Simplify √10

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
C2

Simplify √14

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
C3

Simplify √5

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
C4

Simplify √6

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
Apply

Apply (4)

A1

Simplify √24

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
A2

Simplify √6

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
A3

Simplify √32

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
A4

Simplify √10

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
Extension

Extension (4)

E1

Simplify √3

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
E2

Simplify √75

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
E3

Simplify √72

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.
E4

Simplify √6

Need a hint?
  • Check 4, 9, 16, 25 — does any divide it?
  • None? it is already simplest.