Finish the job
If you take out a small square first, the radicand may still have a square factor — keep going until it is square-free.
Why it works
√72 = 6√2.
Square Roots · Simplify radicals · 3.4
Bigger radicals.
Larger radicands work the same way — just find the largest perfect square. √72 = √(36·2) = 6√2. If you only spot a smaller square first, simplify again until the radicand is square-free.
See it work.
Watch the method
Bigger radicals
√72 = √36 · √2 = 6√2.
1
Split off the perfect square
72 = 36 × 2, and 36 is a perfect square.
2
Separate the roots
The square root of a product is the product of the roots.
3
Take the perfect-square root out
√36 = 6, which comes out front.
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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The idea underneath.
From easy to tricky.
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
Simplify √50
1
Split off the perfect square
50 = 25 × 2, and 25 is a perfect square. √50 = \sqrt25 × 2
2
Separate the roots
The square root of a product is the product of the roots. = √25 · √2
3
Take the perfect-square root out
√25 = 5, which comes out front. = 5√2
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
Where students slip.
Trap
Stopping at 2√18.
Fix
√18 still simplifies; finish to 6√2.
Trap
Arithmetic slips with bigger squares.
Fix
Check: 6² · 2 = 72.
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 √20
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
C2
Simplify √27
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
C3
Simplify √45
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
C4
Simplify √48
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
Apply
Apply (4)
A1
Simplify √24
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
A2
Simplify √45
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
A3
Simplify √24
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
A4
Simplify √28
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
Extension
Extension (4)
E1
Simplify √50
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
E2
Simplify √24
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
E3
Simplify √32
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.
E4
Simplify √12
Need a hint?
- Find the largest perfect-square factor.
- Simplify again if the radicand still has a square factor.