How to Simplify Radicals (Square Roots) Step by Step

To simplify a radical, find the largest perfect square that divides the radicand, split the root using the product property, and take the square root of that perfect square. For example, √50 = √(25 × 2) = 5√2 ≈ 7.0711. A radical is in simplest form when no perfect-square factors (other than 1) remain under the root, no fractions appear inside it, and no radicals sit in a denominator.

Simplifying radicals is one of those skills that looks intimidating and turns out to be mechanical. Every problem uses the same two ideas — perfect-square factors and the product property — and this guide walks through both methods side by side, with hand-verified worked examples at each step.

What Does It Mean to Simplify a Radical?

A radical is “simplified” (in simplest form) when three conditions hold:

No perfect-square factor remains under the root (other than 1). √50 fails this test because 25 hides inside it; 5√2 passes.
No fraction appears under the root. √(4/9) must become 2/3.
No radical appears in a denominator. 1/√2 must become √2/2.

The simplified form is exact in a way decimals never are: 5√2 is √50, while 7.0711 is an approximation. That exactness is why teachers require simplest form — and why rounding early in a multi-step problem quietly corrupts your answer (approximating √2 as 1.41, then multiplying by 5, gives 7.05 instead of the true 7.0711).

Method 1: The Largest Perfect Square Factor

This is the fastest method when you can spot the perfect square. It has four steps.

Step 1: Find the largest perfect square dividing the radicand.
For 50, the perfect-square divisors are 1 and 25. Take 25. (Keep the perfect squares to 15² = 225 handy: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.)

Step 2: Split the radicand into that square times the remainder.
√50 = √(25 × 2).

Step 3: Apply the product property — √(a × b) = √a × √b.
√(25 × 2) = √25 × √2 = 5√2.

Step 4: Add a decimal only if you need one.
5√2 ≈ 5 × 1.41421 = 7.0711.

Another example: √98. Largest perfect-square factor of 98 is 49. √98 = √(49 × 2) = 7√2 ≈ 7 × 1.41421 = 9.8995. If you’d rather watch each step computed for you, Calcrio’s free square root calculator has a Simplify Radicals mode that prints the full factorization working.

Method 2: Prime Factorization (the Factor Tree)

When the largest perfect square isn’t obvious, factor the number completely and let the pairs reveal it. This method never fails.

Worked example: √72.

Factor into primes: 72 = 2 × 2 × 2 × 3 × 3 (or 2³ × 3²).
Group into pairs: (2 × 2) × (3 × 3) × 2 — one 2 is left unpaired.
Move one copy of each pair outside: the pairs contribute 2 × 3 = 6; the leftover 2 stays inside.
Result: √72 = 6√2 ≈ 6 × 1.41421 = 8.4853.

Worked example: √75.

Factor: 75 = 3 × 5 × 5.
Pair: one pair of 5s; the 3 is left over.
Result: √75 = 5√3 ≈ 5 × 1.73205 = 8.6603.

The rule behind both methods is identical — pairs of equal prime factors move outside the root — which is why factoring skill transfers directly. If the factorization step feels rusty, Calcrio’s guide to simplifying fractions uses the same “reduce to simplest form” thinking in a parallel setting, and the square root calculator page shows the factor-tree working automatically.

The Product and Quotient Properties

Both methods rest on two properties worth stating explicitly:

Product property: √(a × b) = √a × √b (for a, b ≥ 0). This is the engine of every simplification above.
Quotient property: √(a/b) = √a / √b (for a ≥ 0, b > 0). Example: √(4/9) = √4 / √9 = 2/3.

The quotient property also handles simplest-form condition #2: a fraction under the root always splits into a root over a root, and then each simplifies separately.

Can You Split a Radical Across Addition?

No — and this is the most expensive mistake in the chapter. √(a + b) ≠ √a + √b. Test it with a = 16, b = 9: √(16 + 9) = √25 = 5, but √16 + √9 = 4 + 3 = 7. Not equal.

This is the cousin of the famous 15-vs-21 trap (√144 + ∛27 = 15, not 21 — the trap being ∛27 = 9). Both mistakes come from treating roots as if they distribute the way multiplication does. Roots split across multiplication and division only. If you want a one-line habit that prevents both errors: simplify each radical fully first, then combine — never reach inside a sum.

How Do You Simplify a Cube Root?

Same idea, one change: group the prime factors into triples instead of pairs, because ∛ asks about three equal factors.

Worked example: ∛54.

Factor: 54 = 2 × 3 × 3 × 3.
Group into triples: one triple of 3s; the 2 is left over.
Move one copy of each triple outside: ∛54 = 3∛2 ≈ 3 × 1.25992 = 3.7798.

And just as squaring checks a square-root simplification, cubing checks a cube-root one: (3∛2)³ = 27 × 2 = 54. ✓

How Do You Add Radicals?

You can only combine like radicals — terms with the same radicand, exactly as you combine like terms in algebra. The trick is that unlike-looking radicals are often like radicals in disguise:

√50 + √18 = 5√2 + 3√2 = 8√2

Simplify each term first (that’s the “simplify each radical fully first” habit), then add the coefficients. If the radicands still differ after simplifying — √2 + √3, say — the expression is already in simplest form and cannot be combined.

Rationalizing the Denominator

Simplest-form condition #3: no radical in a denominator. Fix it by multiplying top and bottom by the radical:

1/√2 = (1 × √2) / (√2 × √2) = √2/2 ≈ 0.70711

The value doesn’t change (you multiplied by √2/√2 = 1); the form just meets the standard. The same move works for cube roots — multiply by whatever completes the triple.

Worked Examples at a Glance

Problem	Method	Simplified	Decimal	Check
√50	largest square: 25	5√2	≈ 7.0711	(5√2)² = 50 ✓
√72	factor tree: 2³ × 3²	6√2	≈ 8.4853	(6√2)² = 72 ✓
√98	largest square: 49	7√2	≈ 9.8995	(7√2)² = 98 ✓
√75	factor tree: 3 × 5²	5√3	≈ 8.6603	(5√3)² = 75 ✓
∛54	triples: 3³ × 2	3∛2	≈ 3.7798	(3∛2)³ = 54 ✓
√50 + √18	like radicals	8√2	≈ 11.3137	5√2 + 3√2 ✓

Notice the check column: squaring (or cubing) your simplified form must return the original radicand. Make that your closing habit on every problem and simplification errors become nearly impossible.

Frequently Asked Questions

What does it mean to simplify a radical?
To rewrite it so no perfect-square factors remain under the root, no fractions appear inside it, and no radicals sit in a denominator. √50 simplifies to 5√2; 5√2 is simplest form.

How do you find the largest perfect square factor?
List the perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …) and take the biggest one that divides the radicand evenly. For 50, that’s 25; for 98, it’s 49.

Can you split a radical across addition?
No. √(a + b) ≠ √a + √b — with a = 16, b = 9, the left side is 5 and the right side is 7. Roots split across multiplication (√(a × b) = √a × √b) and division only.

How do you simplify a cube root?
Group prime factors into triples instead of pairs and move one copy of each triple outside. ∛54 = ∛(27 × 2) = 3∛2 ≈ 3.7798.

How do you add square roots like √50 + √18?
Simplify each to simplest form first, then combine like radicals: 5√2 + 3√2 = 8√2 ≈ 11.3137. Radicals with different radicands can’t be combined.

Why is simplest radical form better than a decimal?
It’s exact — 5√2 is √50, while 7.0711 is rounded. Keeping the exact form through multi-step working avoids compounding rounding errors, and most classes require it.

Keep Practicing

Simplification is a reps skill: ten problems with the squaring-check habit, and the method becomes automatic. For instant step-by-step verification of each factorization, use the free simplify radicals mode on Calcrio’s square root calculator — and if the exponent side of the relationship ever feels shaky, Calcrio’s exponents guide covers the powers that roots undo.