1. Simplifying Radical Expressions

Not all numbers have a nice even square root. For example, if we found on our calculator, the answer would be 2.828427124746190097603377448419... and even this number is a rounded approximation of the square root. Decimal approximations are good estimates, but you may need the exact value. To do this, we will simplify radical expressions using the product property of square roots:

formula

Product Property of Square Roots

We can use the product rule to simplify an expression such as √180 by splitting it into two roots, √36 . √5, and simplifying the first root, 6√5. The trick in this process is being able to translate a problem like √180 into √36 • √5. There are several ways this can be done.

hint

The most common and, with a bit of practice, the fastest method, is to find perfect squares that divide evenly into the radicand, or number under the radical. This is shown in the next example.

		75 is divisible by 25, a perfect square
		Split into factors
		Product rule, take the square root of 25
		Our Solution

If there is a coefficient in front of the radical to begin with, the problem merely becomes a big multiplication problem.

		63 is divisible by 9, a perfect square
		Split into factors
		Product rule, take the square root of 9
		Multiply coefficients
		Our Solution

As we simplify radicals using this method it is important to be sure our final answer can be simplified no more.

		72 is divisible by 9, a perfect square
		Split into factors
		Product rule, take the square root of 9
		But 8 is also divisible by a perfect square, 4
		Split into factors
		Product rule, take the square root of 4
		Multiply

The previous example could have been done in fewer steps if we had noticed that , but often the time it takes to discover the larger perfect square is more than it would take to simplify in several steps.

We can simplify higher roots in much the same way we simplified square roots, using the product property of radicals.

formula

Product Property of Radicals

Often we are not as familiar with higher powers as we are with squares. It is important to remember what index we are working with as we try and work our way to the solution.

		We are working with a cubed root, want third powers
		Test 2, , 54 is not divisible by 8
		Test 3, , 54 is divisible by 27!
		Write as factors
		Product rule, take cubed root of 27
		Our Solution

Just as with square roots, if we have a coefficient, we multiply the new coefficients together.

		We are working with a fourth root, want fourth powers
		Test 2, , 48 is divisible by 16!
		Write as factors
		Product rule, take fourth root of 16
		Multiply coefficients
		Our Solution