Skill: radicals

Simplifying Radicals

A radical (√) asks “what number squared gives this?” — it is the inverse of squaring.
Simplifying means pulling out every perfect square hiding inside: √72 = 6√2.
Learn to hunt perfect squares, combine like radicals, and rationalize denominators.

1 Understand: What a Radical Is

The symbol √ undoes squaring. √49 = 7 because 7² = 49. A simplified radical has no perfect-square
factors left under the symbol and no radical in any denominator.

√72 = √36·2 = 6√2

The three moves

Simplify: find the largest perfect square dividing the radicand, pull it out — √72 = 6√2
Add / subtract: like radicals (same radicand) combine like like-terms — 3√5 + 7√5 = 10√5
Rationalize: never leave a radical in a denominator — 5/√2 = (5√2)/2

Why it matters

Radicals appear in the Pythagorean theorem, the quadratic formula, distance, and every exact answer in geometry.
Simplified form (6√2, not √72) is the standard mathematical handwriting.

Where it’s used

Diagonal distances · the quadratic formula · physics formulas · exact geometry answers.

FiboMap node: alg.radicals · CCSS 8.EE.2, HSN.RN.2 · Know first: exponent rules

2 See It

Perfect squares hide inside radicands. Pull them out and the radical gets simpler.

√50 hides a perfect square: 25. Pull it out as 5, and √2 stays under the radical. 5√2 is the same length, written properly.

Hunt perfect squares (4, 9, 16, 25, 36, 49…) in the factor tree. Each one you find walks out of the radical as its square root: 9 walks out as 3, and 2·3 = 6.

3 Worked Examples

Follow each step. Hunt the perfect square first, then pull it out.

Example 1 Simplify: √72

Thinking: find the largest perfect square dividing 72: 36. Split and pull it out.
√72 = √(36·2) = √36 · √2 = 6√2.
Check: (6√2)² = 36 · 2 = 72.

√72 = 6√2

Example 2 Add: 3√5 + 7√5

Thinking: same radicand → add the coefficients like like-terms.
3√5 + 7√5 = (3 + 7)√5 = 10√5.
Check: it is 3 of something plus 7 of something = 10 of something.

3√5 + 7√5 = 10√5

Example 3 Multiply: √6 · √10

Thinking: multiply radicands, then simplify: √60 = √(4·15).
√60 = 2√15.
Check: (2√15)² = 4 · 15 = 60.

√6 · √10 = 2√15

Example 4 Rationalize: 5/√2

Thinking: multiply top and bottom by √2 to clear the radical from the denominator.
5/√2 = (5√2)/(√2·√2) = 5√2/2 = (5√2)/2.
Check: (5√2/2) · √2 = 5·2/2 = 5.

5/√2 = (5√2)/2

4 Common Mistakes

Three errors show up on almost every radicals quiz. Spot them now and they will never cost you points.

Mistake 1: √9 + √16 = √25 = 5

Wrong

Adding under one radical: √9 + √16 ≠ √(9 + 16).
Radicals do not combine that way.

Right

√9 + √16 = 3 + 4 = 7
Simplify each radical first, then combine only if the radicands match.

Rule: √a + √b ≠ √(a+b). Radicals add like like-terms only.

Mistake 2: simplifying 5√2 to 10

Wrong

“√2 is about 2, so 5 · 2 = 10.”
√2 is irrational (≈ 1.414), not 2.

Right

5√2 stays 5√2
Exact form is the goal — never “finish” a radical by rounding it away.

Rule: 5√2 is already finished. Do not approximate it away.

Mistake 3: leaving 1/√3 as the final answer

Wrong

Stopping with a radical in the denominator.
Standard form forbids radicals “downstairs.”

Right

1/√3 = √3/√9 = √3/3
Multiply top and bottom by √3.

Memory hook: “No radicals downstairs” — rationalize before you box the answer.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Simplify: √98.

Answer

Largest perfect square dividing 98 is 49: √98 = √(49·2) = 7√2.
Check: (7√2)² = 49 · 2 = 98.

2. Simplify: 2√3 + 5√3.

Answer

Same radicand → add coefficients: (2 + 5)√3 = 7√3.

3. Rationalize: 3/√3.

Answer

Multiply by √3/√3: (3√3)/3 = √3 (the 3s cancel).
Either (3√3)/3 or √3 is fully correct.

Key Points to Remember

√a·b = √a · √b — split at the largest perfect square
Like radicals combine: a√m + b√m = (a+b)√m
√a + √b ≠ √(a+b) — simplify first, then combine
Never leave a radical in a denominator — rationalize
√2 ≈ 1.414 is irrational: 5√2 stays 5√2

Practice radical problems
Try the radical tool
See radicals on FiboMap