Square Roots · Capstone · C1

Pick the right move.

The final skill is choosing: if the radicand is a perfect square, evaluate it; if it has a perfect-square factor, simplify to a√b; if it’s square-free, leave it. And you can always estimate.

01 / THE IDEA

See it work.

Watch the method

Pick the right move

Read the radicand, then evaluate, simplify, or leave it.

1 / 3
50\sqrt{50}
1

Split off the perfect square

50 = 25 × 2, and 25 is a perfect square.

50=25×2\sqrt{50} = \sqrt{25 \times 2}
2

Separate the roots

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

=25⋅2= \sqrt{25} \cdot \sqrt{2}
3

Take the perfect-square root out

√25 = 5, which comes out front.

=52= 5\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
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— / WHY IT WORKS

The idea underneath.

The decision

Check the radicand: perfect square → whole-number root; has a square factor → pull it out; square-free → already simplest.

Why it works

Match the move to the radicand.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

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

EX 2medium

Simplify √32

1

Split off the perfect square

32 = 16 × 2, and 16 is a perfect square. √32 = \sqrt16 × 2

2

Separate the roots

The square root of a product is the product of the roots. = √16 · √2

3

Take the perfect-square root out

√16 = 4, which comes out front. = 4√2

EX 3hard

Simplify √32

1

Split off the perfect square

32 = 16 × 2, and 16 is a perfect square. √32 = \sqrt16 × 2

2

Separate the roots

The square root of a product is the product of the roots. = √16 · √2

3

Take the perfect-square root out

√16 = 4, which comes out front. = 4√2

04 / COMMON TRAPS

Where students slip.

Trap

Leaving √50 unsimplified.

Fix

50 = 25·2 → 5√2.

Trap

Simplifying a square-free radical.

Fix

√15 is already simplest.

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 √64

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
C2

Simplify √49

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
C3

Simplify √3

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
C4

Simplify √75

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
Apply

Apply (4)

A1

Simplify √28

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
A2

Simplify √20

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
A3

Simplify √4

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
A4

Simplify √3

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
Extension

Extension (4)

E1

Simplify √45

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
E2

Simplify √75

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
E3

Simplify √50

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.
E4

Simplify √22

Need a hint?
  • Is it a perfect square? a square factor? square-free?
  • Apply the matching move.