To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite 15 as 3⋅5. We can also use the product rule to express the product of multiple radical expressions as a single radical expression.
A General Note: The Product Rule for Simplifying Square Roots
If a and b are nonnegative, the square root of the product ab is equal to the product of the square roots of a and b.
ab=a⋅b
How To: Given a square root radical expression, use the product rule to simplify it.
Factor any perfect squares from the radicand.
Write the radical expression as a product of radical expressions.
Simplify.
Example 2: Using the Product Rule to Simplify Square Roots
Simplify the radical expression.
300
162a5b4
Solution
100⋅3100⋅3103Factor perfect square from radicand.Write radical expression as product of radical expressions.Simplify.
81a4b4⋅2a81a4b4⋅2a9a2b22aFactor perfect square from radicand.Write radical expression as product of radical expressions.Simplify.
Try It 2
Simplify 50x2y3z.
Solution
How To: Given the product of multiple radical expressions, use the product rule to combine them into one radical expression.
Express the product of multiple radical expressions as a single radical expression.
Simplify.
Example 3: Using the Product Rule to Simplify the Product of Multiple Square Roots
Simplify the radical expression.
12⋅3
Solution
12⋅3366Express the product as a single radical expression.Simplify.
Try It 3
Simplify 50x⋅2x assuming x>0.
Solution
Using the Quotient Rule to Simplify Square Roots
Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite 25 as 25.
A General Note: The Quotient Rule for Simplifying Square Roots
The square root of the quotient ba is equal to the quotient of the square roots of a and b, where b=0.
ba=ba
How To: Given a radical expression, use the quotient rule to simplify it.
Write the radical expression as the quotient of two radical expressions.
Simplify the numerator and denominator.
Example 4: Using the Quotient Rule to Simplify Square Roots
Simplify the radical expression.
365
Solution
36565Write as quotient of two radical expressions.Simplify denominator.
Try It 4
Simplify 9y42x2.
Solution
Example 5: Using the Quotient Rule to Simplify an Expression with Two Square Roots
Simplify the radical expression.
26x7y234x11y
Solution
26x7y234x11y9x43x2Combine numerator and denominator into one radical expression.Simplify fraction.Simplify square root.
Try It 5
Simplify 3a4b59a5b14.
Solution
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College Algebra.Provided by: OpenStaxAuthored by: OpenStax College Algebra.Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface.License: CC BY: Attribution.