Simplifying Radicals Calculator

What this calculator does

So, you have a radical expression and you want it simplified, kept in exact form rather than turned into a decimal. This tool does that. It works with expressions of the shape a coefficient times an nth root, and it can simplify a single one, add two together, or multiply and divide two of them.

A dropdown chooses the type of expression, and the input boxes adjust for the values you need.

How to use it

1. Choose the expression type: basic, sum, product, or quotient.
2. Enter the coefficient, radicand, and index for one radical, and a second set for the sum, product, and quotient modes.
3. Press Calculate.

What simplifying a radical means

Simplifying a radical does not mean turning it into a decimal. It means rewriting it in the cleanest exact form, with as small a number as possible left under the root sign. The square root of 8, for example, is not left as it is: since 8 is 4 times 2, and 4 is a perfect square with root 2, the whole thing becomes 2 times the square root of 2. Same value, tidier form. This tool works this way throughout, keeping answers exact and moving whatever it can out from under the radical.

The basic mode: pulling factors out

In basic mode, the tool takes a single radical, a coefficient times the nth root of a radicand, and simplifies it. It looks inside the radicand for a factor that is a perfect nth power, a number whose nth root is whole. When it finds one, it takes that root out of the radical and multiplies it into the coefficient, leaving the rest under the sign. Pulling a perfect square out of a square root, or a perfect cube out of a cube root, is the core move of radical simplification, and this mode does exactly that.

Sums: combining like radicals

Adding radicals has a catch that adding ordinary numbers does not. Two radicals can only be added into one if they are like radicals, meaning they share the same index and the same radicand after simplifying. When that holds, you simply add their coefficients, just as 2 of something plus 5 of something makes 7 of it: 2 times the square root of 3 plus 5 times the square root of 3 is 7 times the square root of 3. When the radicals are not alike, they cannot be merged, and the tool simplifies each and leaves them side by side. This mirrors how you can add like terms in algebra but not unlike ones.

Products and quotients

Multiplying and dividing radicals is more forgiving than adding them, because roots combine cleanly under multiplication. In product mode the tool multiplies the coefficients together, brings the radicands under a common root, and then simplifies the result. In quotient mode it divides the coefficients and combines the radicands the same way before simplifying. These modes let you handle expressions where two radical terms are multiplied or divided, returning a single simplified radical rather than a tangle of separate roots.

A worked example

In basic mode, enter a coefficient of 1, a radicand of 8, and an index of 2. The tool finds the perfect square 4 inside 8, takes its root of 2 outside, and returns 2 times the square root of 2. In sum mode, enter 2 times the square root of 3 and 5 times the square root of 3: since these are like radicals, it combines them into 7 times the square root of 3.

Questions people ask

What does simplifying a radical mean?

Rewriting it in exact form with the smallest possible number under the root sign, by pulling out perfect power factors. The square root of 8 becomes 2 times the square root of 2.

How does the basic mode work?

It finds a perfect nth power factor inside the radicand, takes its root outside the sign, and multiplies it into the coefficient.

Why can't all radicals be added?

Only like radicals, with the same index and radicand, can be combined by adding coefficients. Unlike radicals stay separate, just like unlike terms in algebra.

Are products easier than sums?

Yes. Radicals combine cleanly under multiplication and division, so those modes always return a single simplified radical.

Does it give a decimal answer?

No. It keeps the answer exact, in radical form. For a decimal value of a single root, use the radical or root calculator.

References

On simplifying radicals. A radical is simplified by removing perfect power factors from the radicand, and like radicals, sharing index and radicand, can be combined by adding coefficients.

1. Christopher Stover and Eric W. Weisstein, "Radical," from MathWorld, a Wolfram resource, on radicals and roots as fractional exponents.
2. "Nth Roots and Rational Exponents," College Algebra, Lumen Learning, on simplifying and combining radical expressions.