# How to solve radical expressions with variables and exponents

Simplifying rational exponent expressions: So this is going to be equal to V. Taking the square root of the square is in fact the technical definition of the absolute value.

And it really just comes out of the exponent properties. So let's apply that over here. This expression over here is going to be the same thing as the principal root-- it's hard to write a radical sign that big-- the principal root of 60x squared y over 48x.

## Simplifying / Multiplying Radicals

And then we can first look at the coefficients of each of these expressions and try to simplify that. Both the numerator and the denominator is divisible by Both the numerator and the denominator are divisible by x. Anything we divide the numerator by, we have to divide the denominator by.

Writing out the complete factorization would be a bore, so I'll just use what I know about powers. The r 18 has nine pairs of r ' s; the s is unpaired; and the t 21 has ten pairs of t 's, with one t left over. Your textbook may tell you to "assume all variables are positive" when you simplify. Because the square root of the square of a negative number is not the original number.

You plugged in a negative and ended up with a positive. We're applying a process that results in our getting the same numerical value, but it's always positive or at least non-negative. Taking the square root of the square is in fact the technical definition of the absolute value. But this technicality can cause difficulties if you're working with values of unknown sign; that is, with variables.

You can't know, because you don't know the sign of x itself — unless they specify that you should "assume all variables are positive", or at least non-negative which means "positive or zero". The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.

Simplifying multiplied radicals is pretty simple, being barely different from the simplifications that we've already done. Answer D contains a problem and answer pair that is incorrect.

This problem does not contain any errors. The two radicals that are being multiplied have the same root 3so they can be multiplied together underneath the same radical sign. The two radicals have different roots, so you cannot multiply the product of the radicands and put it under the same radical sign.

### Simplify radical expressions

So, this problem and answer pair is incorrect. You can use the same ideas to help you figure out how to simplify and divide radical expressions.

Recall that the Product Raised to a Power Rule states that. Well, what if you are dealing with a quotient instead of a product?

There is a rule for that, too. The Quotient Raised to a Power Rule states that. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well: A Quotient Raised to a Power Rule. As you did with multiplication, you will start with some examples featuring integers before moving on to more complex expressions like.

Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors. Identify and pull out perfect squares. Rewrite using the Quotient Raised to a Power Rule.

Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors. Identify and pull out perfect cubes. You can simplify this expression even further by looking for common factors in the numerator and denominator. Rewrite the numerator as a product of factors. Identify factors of 1, and simplify.

The answer can't be negative and x and y can't be negative since we then wouldn't get a real answer. In the same way we know that. These properties can be used to simplify radical expressions. A radical expression is said to be in its simplest form if there are. If the denominator is not a perfect square you can rationalize the denominator by multiplying the expression by an appropriate form of 1 e.