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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 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

summary
We can use the product or quotient properties to combine or break down radicands through multiplication or division and this will help us simplify radical expressions. When we're combining two radicals into one using the product or quotient property, the index of the radicals must be the same.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html

Formulas to Know   