Square Roots · Simplify radicals · 3.3

Simplify a radical.

Put it together: find the largest perfect-square factor, split with the product property, and take its root out front. √18 = √(9·2) = 3√2.

01 / THE IDEA

See it work.

The area model

A square root is a side length.

A square with area 18 has side √18.

area 18
side = √18
area = 18
= side × side

Watch the method

Simplify a radical

√18 = √9 · √2 = 3√2.

1 / 3
18\sqrt{18}
1

Split off the perfect square

18 = 9 × 2, and 9 is a perfect square.

18=9×2\sqrt{18} = \sqrt{9 \times 2}
2

Separate the roots

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

=9⋅2= \sqrt{9} \cdot \sqrt{2}
3

Take the perfect-square root out

√9 = 3, which comes out front.

=32= 3\sqrt{2}
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).

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

The full move

Largest perfect-square factor, split, simplify. The leftover (square-free) part stays under the radical.

Why it works

√18 → 3√2.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify √20

1

Split off the perfect square

20 = 4 × 5, and 4 is a perfect square. √20 = \sqrt4 × 5

2

Separate the roots

The square root of a product is the product of the roots. = √4 · √5

3

Take the perfect-square root out

√4 = 2, which comes out front. = 2√5

EX 2medium

Simplify √98

1

Split off the perfect square

98 = 49 × 2, and 49 is a perfect square. √98 = \sqrt49 × 2

2

Separate the roots

The square root of a product is the product of the roots. = √49 · √2

3

Take the perfect-square root out

√49 = 7, which comes out front. = 7√2

EX 3hard

Simplify √8

1

Split off the perfect square

8 = 4 × 2, and 4 is a perfect square. √8 = \sqrt4 × 2

2

Separate the roots

The square root of a product is the product of the roots. = √4 · √2

3

Take the perfect-square root out

√4 = 2, which comes out front. = 2√2

04 / COMMON TRAPS

Where students slip.

Trap

Stopping at √(9·2).

Fix

Finish: √9 = 3 comes out → 3√2.

Trap

Choosing a non-largest square (leaving it unsimplified).

Fix

Use the largest square so the radicand is square-free.

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 √18

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
C2

Simplify √45

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
C3

Simplify √28

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
C4

Simplify √28

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
Apply

Apply (4)

A1

Simplify √12

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
A2

Simplify √12

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
A3

Simplify √28

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
A4

Simplify √8

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
Extension

Extension (4)

E1

Simplify √28

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
E2

Simplify √12

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
E3

Simplify √45

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.
E4

Simplify √12

Need a hint?
  • Find the largest square factor.
  • Take its root out; leave the rest under √.