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Simplifying Radical Expressions

Author: Sophia

what's covered
In this lesson, you will learn how to simplify a given radical expression. Specifically, this lesson will cover:

Table of Contents

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 to know
Product Property of Square Roots
square root of a b end root space equals space square root of a space times space root index blank of b

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

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.

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.

EXAMPLE

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.

EXAMPLE

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.

1a. Product Property

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

formula to know
Product Property of Radicals
m-th root of a b end root equalsm-th root of a times 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.

EXAMPLE

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.

EXAMPLE

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

1b. Quotient Property

Now let's talk about the quotient property of radicals, which states that if you have the nth root of a and you divide that by the nth root of b, you can write this as a single quotient, a over b, and then take the nth root of that. Notice again that our roots must be the same.

formula to know
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

EXAMPLE

fraction numerator square root of 18 over denominator square root of 2 end fraction Use quotient property of radicals
square root of 18 over 2 end root Divide 18 by 2
square root of 9 Simplify
3 Our Solution

Let's try another example going in the other direction.

EXAMPLE

cube root of 27 over 8 end root Use quotient property of radicals
fraction numerator cube root of 27 over denominator cube root of 8 end fraction Since 27 and 8 are both perfect cubes, take the cubed root of the numerator and denominator
3 over 2 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 www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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