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Simplifying Radical Expressions

Simplifying Radical Expressions

Author: Sophia Tutorial
Description:

Simplify a given radical expression.

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  1. Simplifying Radical Expressions

1. Simplifying Radical Expressions

Not all numbers have a nice even square root. For example, if we found square root of 8 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
square root of a b end root space equals space square root of a space • space space root index blank of b

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.

square root of 75
75 is divisible by 25, a perfect square
square root of 25 times 3 end root
Split into factors
square root of 25 times square root of 3
Product rule, take the square root of 25
5 square root of 3
Our Solution

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

5 square root of 63
63 is divisible by 9, a perfect square
5 square root of 9 times 7 end root
Split into factors
5 square root of 9 times square root of 7
Product rule, take the square root of 9
5 times 3 square root of 7
Multiply coefficients
15 square root of 7
Our Solution

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

square root of 72
72 is divisible by 9, a perfect square
square root of 9 times 8 end root
Split into factors
square root of 9 times square root of 8
Product rule, take the square root of 9
3 square root of 8
But 8 is also divisible by a perfect square, 4
3 square root of 4 times 2 end root
Split into factors
3 square root of 4 times square root of 2
Product rule, take the square root of 4
3 times 2 square root of 2
Multiply

The previous example could have been done in fewer steps if we had noticed that 76 equals 36 times 2, 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
m-th root of a b end root space equals space space m-th root of a space • space space m-th root of b

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.

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

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

3 fourth root of 48
We are working with a fourth root, want fourth powers
2 to the power of 4 equals 16
Test 2, 2 to the power of 4 equals 16, 48 is divisible by 16!
3 fourth root of 16 times 3 end root
Write as factors
3 fourth root of 16 times fourth root of 3
Product rule, take fourth root of 16
3 times 2 fourth root of 3
Multiply coefficients
6 fourth root of 3
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
Product Property of Radicals

n-th root of a b end root space equals space n-th root of a space times space n-th root of b

Product Property of Square Roots

square root of a b end root equals square root of a times square root of b

Quotient Property of Radicals

n-th root of a over b end root equals fraction numerator n-th root of a over denominator n-th root of b end fraction