e.g., 3 * ³√16 = ?
Expression
basic radical
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b
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# Simplifying Radicals Calculator

By Maciej Kowalski, PhD candidate
Last updated: Nov 27, 2020
Table of contents:

Welcome to Omni's simplifying radicals calculator, where we'll take on the most common expressions that include radicals. With the power of magic (i.e., mathematics), we'll rewrite them in an easier form. For instance, the basic case here would be simplifying square roots, that is, values of the form `a√b`. However, the expressions can be more complicated, so we'll describe how to simplify radicals in general.

Whichever root expression you're dealing with, this simplest radical form calculator will be able to deal with it!

## Exponents and roots

Whenever we multiply by the same number several times, we can save ourselves some time (time is money, after all), and instead of repeating the multiplication, write the whole thing using exponents. For example:

`5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 = 5⁸`

The small number in the superscript tells us how many times we multiply the big number - in this case; we have eight `5`s.

Taking the root (also called radical) is the inverse operation to the above. In other words, while the exponent turns `5` into `5⁸`, the (eighth) root makes `5⁸` into `5`.

`⁸√(5⁸) = 5`

Let us take this opportunity to mention a couple of essential rules that govern Omni's simplify radicals calculator.

• The order of a radical must be an integer greater or equal to `2`.
• The number under the root must be a non-negative integer. We're aware that radicals of odd order also apply to negative numbers and that rational numbers have roots as well, but, for simplicity, we limit ourselves to the non-negative integer case.

Of course, taking the root is not always that simple. After all, sometimes the number we get is not written as `5⁸` but rather as `390,625`. How do we see that the result is `5` from such a big and complicated number? Or what do we do if it's `390,624` instead? What could that monstrosity be?

Well, most probably, we use some external tools for more complicated tasks - something like our simplify radical expressions calculator. Nevertheless, there are some nifty tricks that we can use, and you bet we will show you all of them! Let's first see how to simplify square roots.

## How to simplify square roots

Square roots are radicals of order `2`. That means that they are the inverse operation to taking the second power (i.e., the square) of a number. For instance, we know that `5² = 25`, so the square root of `25` is `√25 = 5`. Let us focus on such expressions for the remainder of this section, so for now, you can consider our tool as a simplify square roots calculator.

The `25` was easy, but what is, say, `√288`? On the one hand, we have `16² = 256`, and on the other, `17² = 289`, so `√288` should be somewhere between `16` and `17`. Well, sometimes, instead of approximating the result, it's better to transform it a little. That's why we'll describe how to simplify square roots and write `√288` in a prettier way.

The main tool for simplifying radical expressions is prime factorization. Arguably, it's the safest way to deal with such problems since it's fairly easy and always gives the answer. Also, it is the algorithm that our simplifying radicals calculator uses.

We write the number under the radical as a product of prime numbers:

`288 = 2 * 2 * 2 * 2 * 2 * 3 * 3 = 2⁵ * 3²`.

Next, we recall that the root at hand is of order `2`, so we check how many pairs of the same primes we obtained in the factorization: we have two pairs of `2`s, a pair of `3`s, and a single `2` is left alone. Then, we pull the numbers representing pairs in front of the root, and all the singles stay inside. In our case, this gives:

`√288 = √(2⁵ * 3²) = 2 * 2 * 3 * √2 = 12√2`.

That is the best expression we can hope for, and it's precisely what the simplify radicals calculator returns.

We mentioned that simplifying square roots of the form `a√b` is the easiest task there is when dealing with root expressions. But what if the radical's order is higher, say, it's a cube root? Or if we want to add two similar values, i.e., find `a√b + c√d`? Can the tool formerly known as the simplify square roots calculator be turned into a simplify radical expressions calculator with more complicated operations?

Oh, you bet!

## How to simplify radical expressions

The good news is: prime factorization is still our main tool. The bad news is... Actually, there's no bad news. There's just the slightly-less-good news: we're going to play around a bit using root and exponent properties.

In the simplest radical form calculator, you can see four options. We will shortly describe how to simplify the radical expressions given by each of them.

1. `a * ⁿ√b`
For `n = 2` this boils down to simplifying square roots (described in the above section). For radicals of a different order, we repeat the whole reasoning, but instead of pairing the numbers in the prime factorization, we search for groups of the same `n` primes. We pull out the number each such group represents in front of the root, and the ones that didn't form full `n`-tuples stay inside.

2. `a * ⁿ√b + c * ᵐ√d`
The most important rule here is that we can only add radicals with the same order and number under the root symbol, i.e., we must have `n = m` and `b = d`. Otherwise, we can't add the two. There is, however, a catch: a priori, the two summands don't have to be in their simplest radical form. That means that first, we need to apply the reasoning from point 1. above to each of them, and only then check if we can add them up.

3. `a * ⁿ√b * c * ᵐ√d`
As opposed to point 2., here, there's no need to look at each factor separately. In fact, rules of multiplication and the properties of radicals give `a * ⁿ√b * c * ᵐ√d = (a * c) * ᵏ√(bˢ * dᵗ)`, where `k = lcm(n,m)` ( the least common multiple), `s = k / n`, and `t = k / m`. The formula may seem scary, but, in fact, it's really simple in most cases (e.g., when `n = m`). And once we apply it, we get something that returns us to point 1., and simplifying such radical expressions is no biggie.

4. `(a * ⁿ√b) / (c * ᵐ√d)`
Similar to point 3. This time, the rules of mathematics give `(a * ⁿ√b) / (c * ᵐ√d) = (a / (c * d)) * ᵏ√(bˢ * dᵗ)` with `k = lcm(n,m)`, `s = k / n`, and `t = (k * (m - 1)) / m`. The extra `m - 1` factor in `t` comes from rationalizing the denominator. The formula might be a bit more tricky than the previous one, but it again boils down to knowing how to simplify radical expressions from point 1.

Additionally to the basics given above, there is one last thing that may come useful when simplifying radicals. To be precise, after we pull out all that we could out of the root, the root itself can sometimes still be simplified further. If all the primes under the radical have powers with a common factor with the root's order, we can reduce them in a similar way we reduce fractions to obtain an equivalent one.

Symbolically, this rule looks like this:

`k*n√(ak*s * bk*t * ck*u * dk*v * ...) = ⁿ√(aˢ * bᵗ * cᵘ * dᵛ * ...)`.

For instance:

`⁶√(2⁴ * 5¹⁰ * 7²) = ³√(2² * 5⁵ * 7¹)`.

Our simplest radical form calculator uses all these properties to reduce your expression to the easiest possible. Quite a tool, isn't it?

So let's drop the theory and move on to a nice example to see the calculator in action!

## Example: using the simplifying radicals calculator

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 `1`s.)

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 number 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 `2`s), which leaves two `2`s 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 `2`s. 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!

Maciej Kowalski, PhD candidate
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