Square Roots · Simplify radicals · 3.1

Perfect-square factors.

Simplifying starts by finding the largest perfect-square factor of the radicand. For 12, the factors are 1, 2, 3, 4, 6, 12 — and 4 is the largest perfect square, since 12 = 4 × 3.

01 / THE IDEA

See it work.

Watch the method

Perfect-square factors

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

1 / 1
12\sqrt{12}
1

Find a perfect-square factor

4 is a perfect square that divides 12 (12 = 4 × 3).

12=4×312 = 4 \times 3
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number.

streak 0
best 0
… =
— / WHY IT WORKS

The idea underneath.

Spot the square factor

Check perfect squares 4, 9, 16, 25, … against the radicand; use the largest that divides it evenly.

Why it works

12 = 4 × 3 → the 4 is the square.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Largest perfect-square factor of √18

1

Find a perfect-square factor

9 is a perfect square that divides 18 (18 = 9 × 2). 18 = 9 × 2

EX 2medium

Largest perfect-square factor of √75

1

Find a perfect-square factor

25 is a perfect square that divides 75 (75 = 25 × 3). 75 = 25 × 3

EX 3hard

Largest perfect-square factor of √50

1

Find a perfect-square factor

25 is a perfect square that divides 50 (50 = 25 × 2). 50 = 25 × 2

04 / COMMON TRAPS

Where students slip.

Trap

Using a non-square factor.

Fix

You need a perfect-square factor (4, 9, 16, …).

Trap

Using a small square and missing a bigger one.

Fix

Use the largest perfect-square factor for one-step simplifying.

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

Largest perfect-square factor of √98

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
C2

Largest perfect-square factor of √75

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
C3

Largest perfect-square factor of √75

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
C4

Largest perfect-square factor of √72

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
Apply

Apply (4)

A1

Largest perfect-square factor of √45

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
A2

Largest perfect-square factor of √72

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
A3

Largest perfect-square factor of √20

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
A4

Largest perfect-square factor of √20

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
Extension

Extension (4)

E1

Largest perfect-square factor of √28

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
E2

Largest perfect-square factor of √28

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
E3

Largest perfect-square factor of √48

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.
E4

Largest perfect-square factor of √27

Need a hint?
  • Test 4, 9, 16, 25 against the radicand.
  • Pick the largest that divides evenly.