The properties of exponents, which we've talked about earlier,
tell us among other things that
We also know that
If we combine these two things then we get the product property
of radicals and the quotient property of radicals. These two
properties tell us that the square root of a product equals the
product of the square roots of the factors.
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
These properties can be used to simplify radical expressions. A
radical expression is said to be in its simplest form if there
no perfect square factors other than 1 in the radicand
no fractions in the radicand and
no radicals appear in the denominator of a fraction.
If the denominator is not a perfect square you can rationalize
the denominator by multiplying the expression by an appropriate
form of 1 e.g.
are called conjugates to each other. The product of two
conjugates is always a rational number which means that you can use
conjugates to rationalize the denominator e.g.
Video lesson: Simplify the radical