2.2 Operations with Radicals

Quick Answer: MA.912.NSO.1.4 tests one operation at a time on two radicals. Master simplification first, because every operation ends with a simplified radical.

Benchmark MA.912.NSO.1.4 on the Florida Algebra 1 EOC asks you to add, subtract, multiply, or divide numerical radicals. The scope is deliberately narrow: a single arithmetic operation involving two square roots or two cube roots. You will not see nested radicals or radicals with variables on this benchmark — those appear in higher-level courses. The skill that makes every operation manageable is simplifying radicals first, so the section begins there.

Simplifying Numerical Radicals

A radical is in simplest form when the radicand has no perfect-square factor larger than 1 (for square roots) or no perfect-cube factor larger than 1 (for cube roots). To simplify, find the largest perfect square or perfect cube that divides the radicand, factor, and take the root of the perfect factor.

The perfect squares to recognize instantly are 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. The perfect cubes are 8, 27, 64, 125, 216, 343, 512, 729, 1000. Memorizing these saves time on the timed test.

A quick strategy: divide the radicand by the largest perfect square that fits. For √72, check whether 72 is divisible by 36. It is (72 = 36 · 2), so √72 = √36 · √2 = 6√2. If not, try 25, then 16, then 9.

Adding and Subtracting Radicals

Radicals behave like variable terms when adding or subtracting. Just as 3x + 5x = 8x, we have 3√7 + 5√7 = 8√7. The radicand acts like the variable — it must match exactly for the radicals to combine. The coefficients add; the radicand stays the same.

If the radicands do not match, simplify each radical first. For example, √50 + √8 = 5√2 + 2√2 = 7√2. The two radicals looked different, but after simplification both contain √2, so they combine.

An expression like √3 + √5 cannot be combined because 3 and 5 share no perfect-square factors and the radicands remain different. The answer stays as √3 + √5. The EOC may present this as a distractor alongside a false combination like √8, which students sometimes write by adding radicands (3 + 5 = 8). Adding radicands is never valid.

Subtraction follows the same rule: 7√5 − 2√5 = 5√5. If the expression is √72 − √18, simplify first: 6√2 − 3√2 = 3√2.

Multiplying Radicals

To multiply two radicals, multiply the coefficients and multiply the radicands, then simplify. The key property is √a · √b = √(ab) (for square roots) and ∛a · ∛b = ∛(ab) (for cube roots).

Example: 2√3 · 4√5. Multiply coefficients: 2 · 4 = 8. Multiply radicands: √3 · √5 = √15. The product is 8√15, already simplified since 15 has no perfect-square factors.

Example: 3√6 · 2√6. Coefficients: 3 · 2 = 6. Radicands: √6 · √6 = √36 = 6. The product is 6 · 6 = 36. Note that √6 · √6 = 6 — the square root of a number times itself equals the number.

Example with simplification: √10 · √14 = √140. The largest perfect-square factor of 140 is 4 (140 = 4 · 35), so √140 = 2√35.

The EOC multiplication items are limited to two radicals at a time. You will not see a product like √2 · √3 · √5 on a single MA.912.NSO.1.4 question.

Dividing Radicals and Rationalizing the Denominator

Division divides the coefficients and divides the radicands: √a / √b = √(a/b). For example, √72 / √2 = √36 = 6.

When a radical remains in the denominator, the EOC expects you to rationalize it. Rationalizing eliminates the radical from the denominator by multiplying the numerator and denominator by the same radical.

Example: 5 / √3. Multiply numerator and denominator by √3: (5 · √3) / (√3 · √3) = 5√3 / 3. The denominator is now 3, a rational number.

Example with a coefficient: 6√7 / 2√5. Divide coefficients: 6/2 = 3. The radical part is √7 / √5 = √(7/5). Rationalize by multiplying numerator and denominator by √5: 3√7 · √5 / (√5 · √5) = 3√35 / 5.

For cube roots, rationalize by multiplying by the cube root that makes the denominator's radicand a perfect cube. Example: 1 / ∛2. Multiply numerator and denominator by ∛4 (because 2 · 4 = 8, a perfect cube): ∛4 / ∛8 = ∛4 / 2.

EOC Traps and Common Errors

1. Adding radicands: √3 + √3 is 2√3, not √6. The coefficients add, never the radicands.
2. Forgetting to simplify before combining: √20 + √45 looks un combinable, but √20 = 2√5 and √45 = 3√5, so the sum is 5√5.
3. Multiplying coefficients into radicands: 2√3 · 3√2 is 6√6, not √36. The 2 and 3 are coefficients that multiply each other, not radicands.
4. Leaving a radical in the denominator: 4/√2 is not simplified. Rationalize to (4√2)/2 = 2√2.
5. Treating cube roots like square roots: ∛16 cannot be simplified using perfect squares; it needs a perfect-cube factor. The largest perfect cube dividing 16 is 8, so ∛16 = ∛(8·2) = 2∛2.

The EOC reference sheet does not include perfect-square or perfect-cube lists, so memorization is essential. Practice until recognizing the largest perfect factor becomes automatic.