Two ways to simplify square roots, plus cube roots, coefficients, rationalizing the denominator and adding like radicals, all with worked examples.
What does it mean to simplify a radical?
Simplifying a radical means rewriting it so the number under the root has no perfect square factors left (no perfect cube factors, for a cube root). √72 simplifies to 6√2, because 72 = 36 × 2 and √36 = 6.
The simplified form has the same value, so why bother? Simplified radicals are easier to compare, combine and check. √8 and √18 look unrelated, but once you write them as 2√2 and 3√2, you can see they are multiples of the same thing.
It also gives everyone the same final answer. 6√2, √72 and 2√18 are all equal, but only 6√2 is in simplest form, so that’s what an answer key will show.
How to simplify radicals with perfect squares
Find the largest perfect square that divides the number, split the root into two roots, and take the square root of the perfect square. This uses the rule √(ab) = √a × √b.
How do you simplify square roots with prime factorization?
To simplify square roots with primes, factor the number completely and circle each pair of equal primes. Each pair comes out of the root as one copy; any prime without a partner stays inside.
This is slower than spotting a perfect square, but it never misses one. Here’s √180:
What do you do with a number already in front of the radical?
Simplify the radical first, then multiply whatever comes out by the coefficient already in front.
For 3√12: √12 = √4 × √3 = 2√3. Then 3 × 2√3 = 6√3.
How do you simplify cube roots?
Simplify a cube root by pulling out perfect cubes (8, 27, 64, 125) instead of perfect squares. With prime factors, you need a group of three equal primes to bring one copy out.
For ∛54: 54 = 27 × 2, and ∛27 = 3, so ∛54 = 3∛2. For ∛16: 16 = 8 × 2, so ∛16 = 2∛2.
How do you simplify a radical with a variable?
Treat the exponent the way you treat prime pairs: every two copies of a variable come out of a square root as one. Assume the variables are not negative, as most textbooks do.
So √(x⁴) = x², because x⁴ is x² × x². For an odd power, split off one copy: √(x³) = √(x² × x) = x√x. Put numbers and variables together and √(50x³) = √(25x²) × √(2x) = 5x√(2x).
The simplify radicals calculator works with numbers, so radicals with variables are a step you’ll do by hand using this pairing rule.
How do you multiply and divide radicals?
Multiply the numbers under the roots, then simplify the result. Division works the same way: divide under one root, then simplify.
For √6 × √3, multiply to get √18, which simplifies to 3√2. Sometimes the product is a perfect square and the root disappears completely: √2 × √8 = √16 = 4.
How do you rationalize the denominator?
To rationalize a denominator, multiply the top and bottom by the radical in the denominator. That turns the bottom into a whole number without changing the value.
For √(1/2), first split it into 1/√2. Multiply top and bottom by √2: the bottom becomes √2 × √2 = 2, so the answer is √2/2.
How do you add and subtract radicals?
You can only add radicals with the same number under the root, called like radicals. Simplify each one first, because two radicals that look different may turn out to be alike.
Take √8 + √18. Neither looks like the other, but √8 = 2√2 and √18 = 3√2. Now they are both multiples of √2, so add the coefficients:
What are the most common mistakes when you simplify radicals?
The biggest mistake is splitting a root over addition. √(9 + 16) is √25 = 5, not √9 + √16 = 7. The product rule works for multiplication only.
Frequently asked questions
How do you rationalize √(1/2)?
Write it as 1/√2, then multiply the top and bottom by √2. The bottom becomes 2, so √(1/2) = √2/2.
How do you simplify √72?
Write 72 as 36 × 2. The square root of 36 is 6, so √72 = 6√2.
What is the fastest way to simplify square roots?
Divide by the largest perfect square you can spot (4, 9, 16, 25, 36, and so on), then take its root. If you can’t spot one, use prime factorization and pull out pairs.
When is a radical fully simplified?
When the number under the root has no perfect square factor other than 1, there are no fractions under the root, and no radical is left in a denominator.
Can you add √2 and √3?
Not into a single radical. You can only combine like radicals, which have the same number under the root. √2 + √3 is already simplified.
Is 2√18 simplified?
No. 18 still has a factor of 9, so 2√18 = 2 × 3√2 = 6√2. Keep going until the number under the root has no square factor.