How to Simplify Radicals/Roots

A radical √n, where n is factorable, can often be simplified to the form x√y, where x and y are both less than n. Simplifying square roots, cube roots, etc. is useful when you need to evaluate a complicated expression involving radicals, and it lets you write radicals in a more compact form.

You can use the calculator on the left to reduce radicals, or apply the method described below.

Fundamental Rule:

If n can be written as x²y, then

√n = √x²y = x√y.

Likewise, if m = a³b and p = c⁴d, then

∛m = ∛a³b = a∛b and

∜p = √c⁴d = c∜d.

As you can see above, the key to simplifying a square root is to break it down into a square times another number. For example, since 12 = (2²)(3), the square root of 12 is 2√3. The same principle applies to reducing a cube root (or fourth root), you must express the radicand (number underneath the radical bar) as the product of a cube (or fourth power) and another number.

If you don't immediately know the complete factorization of a number, you can simplify a root in stages.

Example: Simplify ∛3969000

First, we can see that 3969000 = (1000)(3969) = (10³)(3969). This gives us

∛3969000 = 10∛3969.

Next, we notice that 3969 is highly divisible by 3:

3969/3 = 1323
1323/3 = 441
441/3 = 147
147/3 = 49

Thus, 3969 = (3⁴)(49), or (3³)(147). This gives us

10∛3969 = 30∛147

as the final answer.

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