Simplifying Radical Expressions

Before we can simplify a radical expression, we must know the important properties of radicals.

PRODUCT PROPERTY OF SQUARE ROOTS

For all real numbers a and b,

That is, the square root of the product is the same as the product of the square roots.

There's an analogous quotient property:

For all real numbers a and b, b ≠ 0:

SIMPLIFYING RADICALS

The idea here is to find a perfect square factor of the radicand, write the radicand as a product, and then use the product property to simplify.

Example 1:

Simplify.

9 is a perfect square, which is also a factor of 45.

Use the product property.

If the number under the radical has no perfect square factors, then it cannot be simplified further. For instance the cannot be simplified further because the only factors of 17 or 17 and 1. So, there are no perfect square factors other than 1.

Example 2:

Simplify.

Use the quotient property to write under a single square root sign.

Divide.

If the number in the denominator is not a factor of the number in the numerator, we must rationalize the denominator, that is have the radical sign appear only in the numerator.”

Example 3:

Simplify.

Sometimes we need to use a combination of steps.

Example 4:

Simplify.

21 and 9 have a common factor of 3, so reduce the fraction under the radical.

Now rationalize the denominator.

We can only add or subtract two radical expressions if the radicands are the same. For example, cannot be simplified any further. But we can simplify by using the distributive property, because the radicands are the same.

Be careful! Sometimes, the radicands look different, but it's possible to simplify and get the same radicand.

Example 5:

Simplify.

Simplify both radical expressions:

Now, the radicands are the same. Add using the distributive property.

VARIABLE EXPRESSIONS UNDER THE RADICAL SIGN

When you have variables under the radical sign, see if you can factor out a square.

Example 6:

Simplify.

Factor the radicand as the product of a and a squared expression.

Use the product property of square roots:

Simplify.

BE CAREFUL THAT YOUR VARIABLE ONLY STANDS FOR A POSITIVE NUMBER. For instance, , but if a < 0, then (the opposite of a), since the square root sign always indicates the positive square root. Since there is no way for us to know if a is positive or negative, we use absolute value. So, .

The only time we have to worry is if the index of the radical is even, the exponent of the radicand is even, and the exponent of the root is odd.

Example 7:

Simplify.

Rewrite the radicand using squared expressions where possible.

Simplify. (a could not be negative because we would not have been able to take the square root of a³. However, b could be negative, so use absolute value signs.)

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