Before we can simplify a radical expression, we must know the important properties of radicals.
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:
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.
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.
Use the quotient property to write under a single square root sign.
An expression is considered simplified only if there is no radical sign in the denominator. If we do have a radical sign, we have to rationalize the denominator. This is achieved by multiplying both the numerator and denominator by the radical in the denominator. Note that here, we're just multiplying by a special form of 1, so it doesn't change the value of the expression.
Sometimes we need to use a combination of steps.
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.
Simplify both radicals:
Now, the radicands are the same. Add using the distributive property.
When you have variables under the radical sign, see if you can factor out a square.
Factor the radicand as the product of a and a squared expression.
Use the product property of square roots:
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.
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.)