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1 In order to simplifying radical expression, it s important to understand a few essential properties. Product Property of Like Bases a a = a Multiplication of like bases is equal to the base raised to the sum of the exponents Power to a Power Property (a ) = a A power raised to a power is equal to the base raised to the product of the exponents Product Property of Radicals a b = a b Quotient Property of Radicals The square root of a product is equal to the product of the square roots = The square root of a quotient is equal to the quotient of the square roots Division Property of Equality = = 1 Dividing a square root by itself equals 1. Simplifying Radical Expressions 1. take the square root of the perfect square numbers and/or variables, or 2. take the square root of the largest perfect square factors of the expression, while keeping factors that have no square numbers as their factors under the radical. 3. if a variable expression results from a square root, absolute value bars must enclose the expression to insure no complex numbers result from taking the square root of a negative value. Let s look at a few simple examples before we begin. 4 = 2 = 2 9 = 3 = 3 x = x a = (a ) = a Remember you re looking for the largest perfect square number that is a factor of a. m = (m ) = m Shannon Richards Page 1

2 Note: If you multiply each of the resulting values from above to itself, you will get the expression that is under the radical. Taking the square root of a value and squaring a value are opposite operations, just like adding and subtracting; multiplying and dividing; and factoring and distributing. Now let s look at specific examples that will address what has been discussed. Example Factor 32 into a perfect square number and a non-perfect square number 4 2 Taking the square root of Final simplified answer Note: If the largest perfect square number was not factored out of 32, it can still be simplified, but the process is twice as long. For instance, if a student thought to factor the perfect square number 4 out of 32, below would be the process they would take to arrive at the answer 4 2: Factor 32 into a perfect square number and a non-perfect square number 2 8 Taking the square root of This is not simplified completely since 8 has a factor of 4, a perfect square number; therefore, we must continue to simplify Factor 8 into a perfect square number and a non-perfect square number Taking the square root of Multiply the coefficients to the radical 4 2 Final simplified answer Shannon Richards Page 2

3 Note: The perfect square number 4 was simplified out of the radical first, and then a second time when factoring 8. If 4 was factored out twice, then 4 4, or 16, was the largest perfect square number of 32. I like to call this method the scenic route. Example For this example, we have options. We could a) Simplify each radical first, and then multiply. Keep in mind there may be a need to simplify after multiplying. b) Multiply first, and then simplify. This method may produce large numbers that may be difficult to simplify. a) Factor 8 into a perfect square number and a non-perfect square number Taking the square root of Commutative property of multiplication Factor 44 into a perfect square number and a non-perfect square number Taking the square root of Multiply coefficients; Final simplified answer b) Factor 176 into a perfect square number and a non-perfect square number 4 11 Taking the square root of 16; Final simplified answer Shannon Richards Page 3

4 Note: The second method of multiplying first then simplifying did make the number of steps fewer than the first method; however, for some, finding the largest perfect square root of 176 may be a challenge. In the end, the goal is to determine the simplified radical expression. Example 3 Using the Quotient Property of Radicals Taking the square root of 14; Final simplified answer Note: This is the simplified radical expression, because 17 is a simplified radical. Other similar problems may require further simplification of the radical, which may in turn require simplification between the numerator and the denominator. Example 4 It s important to notice that the numerator is not contained within a radical; therefore, it would be incorrect to apply the Quotient Property of Radicals in this case. This problem requires the denominator to be rationalized, which applies the Division Property of Equality. In this case our version of 1 will be. or Multiplying fractions and applying the Product Property of Radicals Taking the square root of 36 Simplify the coefficient in the numerator with the denominator; Final simplified answer Shannon Richards Page 4

5 Note: For the same reason it would have been incorrect for us to simplify the 2 and the 6 in the initial problems, it would be incorrect for us to simplify the 3 and the 6 in this final version. Both the numerator and the denominator require radicals before the Quotient Property of Radicals can be applied. Example 5 Notice this problem has a radical in both the numerator and the denominator; therefore, the Quotient Property of Radicals can be applied in this case. This has a similar feel as Example 2, in that we have options in how we initiate the simplification process. The value 256 under the radical in the numerator is a perfect square number, but the value 32 in the denominator is not. This says there might be a need to rationalize the denominator at some point along the simplification process. However, notice that 256 is divisible by 32. This makes the problem extremely easy to simplify Using the Quotient Property of Radicals Division Factor 8 into a perfect square number and a non-perfect square number 2 2 Final simplified answer Example 6 a b c Let s break each factor down into square parts and non-square parts. a = a a, where a is the perfect square part b is a perfect square c = c c, where c is the perfect square part a b c = a a b c c Rewriting the expression, separating the square parts and non- square parts using the Product Property of Like Bases a b c a c Shannon Richards Page 5

6 a b c a c a b c ac Taking the square root of perfect squares Including the absolute value bars as indicated in #3 under simplifying radical expressions, as well as eliminating the exponents of 1 on a and c. These are understood to be 1 when they are not explicitly written. Final simplified answer Example 7 Notice the radical 2a cannot be simplified as a radical, nor can it be simplified with the 4 in the numerator. In this instance, we need to rationalize the denominator using. = () Multiplying by 1 to rationalize the denominator Taking the square root of (2a) and including the absolute value bars Simplifying the coefficient in the numerator with the coefficient in the denominator; Final simplified answer Example 8 Notice the fraction under the radical has both numbers and variables, which can be simplified before proceeding with simplifying the radical. Since neither the numerator nor the denominator is a perfect square, simplifying the fraction is the best first step for this problem. Dividing a common factor of (2mp) from the numerator and the denominator Shannon Richards Page 6

7 Using the Quotient Property of Radicals Factor 16m p into a perfect square and a non-perfect square Taking the square root of 16m p Rationalizing the denominator = Taking the square root of 3 ; Final simplified answer Note: Each step above was show in detail; however, with understanding and practice, some of these steps may be merged. This problem could have utilized the rationalizing of the denominator as the first step, but the values under the radical would have gotten large quickly. With that said, so long as the properties are clearly followed, the path taken will lead to the same simplified radical expression. Shannon Richards Page 7

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