A property of mathematics says that square roots can be distributed over multiplication. That means a radical such as can be written as

Or a radical expression such as can be written as which simplifies to

Try these:

The property is often written: or

A related property says that square roots are also distributive over division:

or

Simplify:

Try one more:

Because 75 is not a perfect square... What makes this one different?... we don’t know what its square root is. We need to do something else to simplify it.

One possibility is to get an approximation from a calculator

But another way to simplify it is to use what we just learned about multiplying square roots.

What just happened?What allowed us to change to ?

One of our numbers was the square root of a perfect square.

Since we know the square root of perfect squares, we can write as

A radical term with all perfect square factors extracted from the radical is said to be in Simplest Radical Form.

Look at this example: Is it in simplest radical form?Have all perfect square factors been removed?

No—what is a perfect square factor of 24?

Did you say 4?We need to remove a factor of 4, so we will have: You’re right!

Just like when we simplify fractions, when we simplify radicals, we are not through until all possible factors have been removed.

And just like when we simplify fractions, it saves a step or two if we remove the greatest possible factor first.

Try this one: