Simplifying Roots (extended version)

The of any positive number is the number that can be squared to get the number whose square root we are seeking. For example,

= 416 because if we square 4 we get 16, which is the number whose square root is being found.

= 864 because 2 = 648

= 10100 because 102 = 100

The symbol is called a radical and it is read as “the square root of.”

The number underneath the radical is called the “radicand.” In the expression 36 , the radicand is 36.

It should be noted that each positive number has 2 square roots. One is the positive or principal square root, and the other is the negative square root.

= 24 because 22 = 4

−=− 24 because 2 =− 4)2(

We are usually interested in the positive square root. If we want the negative root, we put a negative sign in front of the radical.

−=− 24 −=− 749

Note that we cannot have a negative sign under the radical.

− 4 is not a , because there is no number that we can multiply by itself and get − 4 . (In MAT 1033 and MAC 1105 you will learn how to deal with this situation.)

This instructional aid was prepared by the Tallahassee Community College Learning Commons. To simplify a radical we must look for and remove any perfect square factors that may be in the radicand. REMEMBER that a perfect square is the square of an .

2 = 93 2 = 164 Perfect 2 = 648

A radical expression is in simplest form if the radicand contains no perfect square factors. To simplify a radical we will first find the prime factorization of the radicand and rewrite the radicand in exponential form.

= 264 6 prime factorization of 64 in exponential form.

If the exponent is an even number, then the number itself is a perfect square.

To take the square root of 26 we remove the radical and divide the exponent by 2.

EXAMPLE:

4 32/66 =⋅⋅==== 82222226

Once we have divided the exponent by 2, we can multiply out the remaining factors.

EXAMPLES:

1 22/44 =⋅==== 9333338

44 22/22/424 =⋅=⋅=⋅=⋅= 12343232321

96 2/22/222 =⋅=⋅=⋅= 147272721

Often the number we wish to simplify is not a perfect square. We then have to find any perfect square factors contained in the number and remove them from under the radical by taking their square roots. To simplify 40 , first find the prime factorization of 40.

3 ⋅= 524 0

NOTICE that the exponents are odd numbers. This means that 23 and 51 are not perfect squares.

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

2 • Any prime factor with an exponent of 1 will not be a perfect square nor will it contain a perfect square.

• Any prime factor with an even exponent will be a perfect square.

• Any prime factor with an odd exponent of 3 or higher will contain a perfect square factor.

3 ⋅= 524 0 23 contains a perfect square 0 ⋅⋅= 5224 112 23 is written as 2 ⋅ 212

The Product Property of Square Roots allows us to rewrite a product under a radical as a product of 2 separate radicals.

0 112 2 ⋅⋅=⋅⋅= 5225224

We now have on radical which is a perfect square and one which is not. We can take the square root of the perfect square and multiply the factors remaining under the radical.

2 =⋅⋅ 102522

The complete process is as follows:

0 3 ⋅= 524 Find the prime factorization of 40 ⋅⋅= 522 112 Rewrite 23 as 22 ⋅ 2 2 ⋅⋅= 522 Separate the perfect squares = 102 Take square roots. The solution is read as “2 times the square root of 10”

EXAMPLE: Simplify 96

6 5 ⋅= 329 Find the prime factorization of 96 ⋅⋅= 322 114 Rewrite 25 as 24 ⋅ 2 4 ⋅⋅= 322 Separate the perfect squares 2/4 ⋅= 62 Take square roots. = 2 62 Simplify 22 to get 4 = 64 The solution is read as “4 times the square root of 6”

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

3 EXAMPLE: Simplify 1805 . Notice that this is “5 times the square root of 180.” We must simplify 180 first and then multiply by 5.

22 ⋅⋅= 53251805 Find the prime factorization of 180 22 ⋅⋅= 5325 Separate the perfect squares 2/22/2 ⋅⋅⋅= 5325 Take square roots. ⋅⋅⋅= 5325 Multiply = 53 0 The solution is read as “30 times the square root of 5”

EXAMPLE: Simplify − 328

−=− 28328 5 Find the prime factorization of 32 5 4 4 ⋅−= 228 Rewrite 2 as 2 ⋅ 2 −= 4 228 Separate the perfect squares 2/4 ⋅⋅−= 228 Take square roots. 2 ⋅⋅−= 228 Simplify 22 to get 4 and multiply −= 23 2 The solution is read as “4 times the square root of 6”

Many of the expressions we will need to simplify will contain variables.

2 = xx 36 because ( 3 ) = xx 6 2 = yy 51 0 because ( 5 ) = yy 10

• Any variable radical expression which has an even exponent will be a perfect square.

2 = xx

• Any variable radical expression which has an exponent of 1 will not be a perfect square nor will it contain a perfect square factor.

= xx

• Any variable expression which has an odd exponent of 3 or higher will contain a perfect square factor.

3 2 /2 2 ==⋅= xxxxxxx

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

4 EXAMPLES:

== xxx 32 /66

18 == xxx 92 /18

9 8 /8 2 ==⋅= 4 xxxxxxx

We often have radicals which have both numbers and variables.

EXAMPLE: Simplify 45 yx 32

45 yx 32 53 ⋅⋅⋅= yx 322 Find the prime factorization of 45 53 222 ⋅⋅⋅⋅= yyx Rewrite y 3 as 2 ⋅ yy 3 ⋅⋅= 222 5yyx Separate the perfect squares and take square roots = 53 yx y

EXAMPLE: Simplify 72 cba 965

72 cba 965 32 ⋅⋅⋅⋅= cba 96523 Find the prime factorization of 72 322 86422 ⋅⋅⋅⋅⋅⋅⋅= ccbaa Rewrite ,2 a 53 , and c 9 32 ⋅⋅⋅⋅= 86422 2accba Separate the perfect squares and take square roots 32 ⋅⋅⋅⋅= 432 2accba Multiply the numbers = 432 26 accba

EXAMPLE: Simplify 273 yxy 34

34 343 73 332 ⋅⋅= yxyyxy Find the prime factorization of 27 242 333 ⋅⋅⋅⋅= yyxy Rewrite 33 and y 3 33 ⋅⋅= 242 3yyxy Separate the perfect squares and take square roots 2 ⋅⋅⋅= 333 yyxy Multiply the numbers and multiply the variables = 22 39 yyx

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

5 EXAMPLE: Simplify 184 yxy 45

45 324184 ⋅⋅⋅= yxyyxy 452 Find the prime factorization of 18 324 42 ⋅⋅⋅⋅= yxxy 4 Rewrite x5 as 4 ⋅ xx 34 ⋅⋅= 442 2xyxy Separate the perfect squares and take square roots ⋅⋅⋅= 22 234 xyxy Multiply the numbers and multiply the variables = 32 21 2xyx

EXERCISES. Simplify each of the following. a) 49 b) 12 c) 40

d) 803 e) x14 f) y15

g) 24 yx 52 h) 203 ba 25 i) 642 yx 108

j) 1202 yxxy 35

KEY a) 7 b) 32 c) 102 d) 51 2 e) x 7 f) 7 yy g) 2 62 yx y h) 2 56 aba i) 16 yx 54 j) 23 3 04 xyyx

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

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