Simplifying Radicals Calculator (2024)

Say that you've only just learned about roots and how to simplify radicals. That doesn't stop your teacher from giving the whole class a task to try to do on your own while they sit behind their desk with a phone in hand. They seem to think simplifying radical expressions will take you long enough for them to browse through the newsfeed. Why don't we prove them wrong?

The task is to find the sum, product, and quotient of 2√6 and ⁴√64. In other words, we need to compute 2√6 + ⁴√64, 2√6 × ⁴√64, and 2√6 / ⁴√64.

Well, how fortunate that we have Omni's simplifying radicals calculator at hand!

Let's start with the sum. In the calculator, we see an option to choose the radical expression that we want to find. We're interested in adding two values, so we choose "sum" under "Expression." That will trigger a symbolic representation of the operation to appear.

next, we need to input the numbers. The simplify radical expressions calculator shows the sum as a × ⁿ√b + c × ^m√d, so for our case, we input

a = 2, b = 6, n = 2, c = 1, d = 64, m = 4.

(note how n = 2 is the default since, usually, we deal with square roots. Also, we didn't really need to input c = 1 – the calculator understands blank fields for a and c as 1s.)

Once we give all the numbers, we can read off the result from underneath the variable fields. Observe how the calculator also gives a step-by-step solution to your problem.

For the product and quotient, we repeat the above steps (the a, b, and so on don't change), but choose the correct option under "Expression" – product and quotient, respectively.

nevertheless, for the horrific times when you can't use the internet, let's see how to simplify the radicals ourselves without our tool's help.

We begin with the sum: 2√6 + ⁴√64. First of all, we need to find prime factorizations of the two numbers under the radicals:

6 = 2 × 3,

64 = 2 × 2 × 2 × 2 × 2 × 2 = 2⁶.

The first root is of order 2, so we need to find pairs of the same number in the factorization. We see that there is none, so that summand is already in its simplest form.

For the second, we need fours. Indeed, there is one such solution (four 2s), which leaves two 2s alone. We pull the numbers representing the groups of four out of the radical and keep the rest inside.

2√6 + ⁴√64 = 2√6 + ⁴√(2⁶) = 2√6 + 2 × ⁴√(2²)

Unfortunately, this is not over yet. In the second summand, we see that the order of the root and the powers of all (i.e., of the only one) numbers under it have a common factor – 2. Therefore, we reduce the two numbers by that factor.

2√6 + ⁴√64 = 2√6 + 2 × ⁴√(2²) = 2√6 + 2√2

Note how we didn't write the ² with the second radical because, by convention, we write square roots without that number.

The two summands have different numbers under the radicals (but the same root orders), so we can't add them – this is the simplest form of the expression.

Now for simplifying the radical expression with the product: 2√6 × ⁴√64. The two roots have orders 2 and 4, respectively, and lcm(2,4) = 4. We follow the instructions given in the above section and get:

2√6 × ⁴√64 = 2 × ⁴√(6² × 64) = 2 × ⁴√2304.

Next, we find the prime factorization of the number under the root:

2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = 2⁸ × 3².

We look for fours of the same prime in the factorization and find two: a couple of fours of 2s. We pull these out of the radical and get

2√6 × ⁴√64 = 2 × ⁴√2304 = 2 × ⁴√(2⁸ × 3²) = 2 × 2 × 2 × ⁴√3² = 8 × ⁴√3².

We have a similar situation to what we had with the sum: the order of the root and the primes' powers under it have a common factor. We reduce them and get our final answer:

2√6 × ⁴√64 = 8 × ⁴√3² = 8√3.

Lastly, let's see how to simplify the radical expression with the quotient: 2√6 / ⁴√64. We recall that lcm(2,4) = 4 and the instructions from the above section to obtain:

2√6 / ⁴√64 = (2 / 64) × ⁴√(6² × 64³) = 0.03125 × ⁴√9,437,184.

We find the prime factorization of the number under the root:

9,437,184 = 2²⁰ × 3²,

and look for fours of the same primes. In this case, we have five fours of 2. We pull these out of the radical and get:

2√6 / ⁴√64 = 0.03125 × ⁴√9,437,184 = 0.03125 × ⁴√(2²⁰ × 3²) = 0.03125 × 2⁵ × ⁴√(3²) = ⁴√(3²).

Again, we can reduce the order of the root and the powers of the primes under it. That gives us a final answer of:

²√6 / ⁴√64 = ⁴√(3²) = √3

Arguably, it took some time to write all the details, but the operations themselves weren't too terrible. Still, it makes us appreciate how much work the simplifying radicals calculator can save us. And after reading through this article, even doing it by hand must take less than the teacher suspects!