Square Roots · Simplify radicals · 3.2

The product property.

The product property splits a root over a product: √(ab) = √a · √b. So √12 = √(4·3) = √4 · √3 = 2√3. This is the engine of simplifying radicals.

01 / THE IDEA

See it work.

Watch the method

The product property

√12 = √4 · √3 = 2√3.

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12\sqrt{12}
1

Split off the perfect square

12 = 4 × 3, and 4 is a perfect square.

12=4×3\sqrt{12} = \sqrt{4 \times 3}
2

Separate the roots

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

=4⋅3= \sqrt{4} \cdot \sqrt{3}
3

Take the perfect-square root out

√4 = 2, which comes out front.

=23= 2\sqrt{3}
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.

Split the root

A square root of a product equals the product of the square roots — so the perfect-square part comes out front.

Why it works

√(4·3) = √4·√3 = 2√3.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Simplify √12

1

Split off the perfect square

12 = 4 × 3, and 4 is a perfect square. √12 = \sqrt4 × 3

2

Separate the roots

The square root of a product is the product of the roots. = √4 · √3

3

Take the perfect-square root out

√4 = 2, which comes out front. = 2√3

EX 2medium

Simplify √12

1

Split off the perfect square

12 = 4 × 3, and 4 is a perfect square. √12 = \sqrt4 × 3

2

Separate the roots

The square root of a product is the product of the roots. = √4 · √3

3

Take the perfect-square root out

√4 = 2, which comes out front. = 2√3

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

Splitting over a sum.

Fix

√(a+b) ≠ √a + √b — only products split.

Trap

Leaving √4 inside.

Fix

Take the perfect-square root (2) out front.

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 √27

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
C2

Simplify √48

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
C3

Simplify √48

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
C4

Simplify √45

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
Apply

Apply (4)

A1

Simplify √45

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
A2

Simplify √45

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
A3

Simplify √48

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
A4

Simplify √50

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
Extension

Extension (4)

E1

Simplify √50

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
E2

Simplify √18

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
E3

Simplify √48

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.
E4

Simplify √75

Need a hint?
  • Write the radicand as (square) × (rest).
  • Take the square root of the square out front.