10­2 Simplifying Radicals radical (noun, adjective) ​
rad ih kul Definition: ​
A radical is the indicated root of a quantity, such as √7 or √X . The sign that indicates a root is to be taken is a radical sign. Word Origin: ​
from the Latin word “radix,” which means “root” Math Usage: ​
A radical expression contains a radical. A radical expression is simplified if the following are true: • The radicand has no perfect­square factors other than 1. • The radicand contains no fractions. • No radicals appear in the denominator of a fraction. Simplified Not Simplified √2
5
x
3√5 9√x 3√12
4
2
√
√
7
Essential Understanding​
You can simplify radical expressions Using multiplication and division properties of square roots. You can use the Multiplication Property of Square Roots to simplify radicals by removing perfect­square factors from the radicand. Simplify each radical expression. A.) √99
B.) − √60
C.) 5√700 Sometimes you can simplify radical expressions that contain variables. A variable with an even exponent is a perfect square. Avariable with an odd exponent is the product of a perfect square and the variable. For example, n3 = n2 • n , so √n3 = √n2 • n . In this lesson, Assume that all variables in radicands represent nonnegative numbers. E.) − 21√27x9 D.) √192s2 √243y F.) − 2
3
.You can use the Multiplication Property of Square Roots to write √a • √b = √ab. Simplify each product. G.) 31 √6 • √24
H.) 5√6 • 61 √216 I.) √18n • √98n3 J.) √2y • √128y5 K.) − 9√28c2 • 31 √63a
(
)
L.) − 31 √18c5 • − 6√8c9 M.) ​
Park​
A park is shaped like a rectangle with a length 5 times its width w. What is a simplified expression for the distance between opposite corners of the park? You can simplify some radical expressions using the following property. When a radicand has a denominator that is a perfect square, it is easier to apply the Division Property of Square Roots first and then simplify the numerator and denominator of the result. When the denominator of a radicand is not a perfect square, it may be easier to simplify the fraction first. Simplify each radical expression. √
N.) 7
6
32
O.) √
3x3
64x2
When a radicand in a denominator is not a perfect square, you may need to rationalize the denominator​
to remove the radical. To do this, multiply the numerator and denominator by the same radical expression. Choose an expression that makes the radicand in the denominator a perfect square. It may be helpful to start by simplifying the original radical in the denominator. Simplify each radical expression. √
P.) 11
49a5
4a3
Q.) √5
√8x
R.) 22
√11
S.) 8√7s
√28s
3