!
Notes 7.5-­‐ Operations with Radical Expressions Why? Golden rectangles have been used by artists and architects to create beautiful designs. Many golden rectangles appear in the Parthenon in Athens, Greece. The ratio of the lengths of the sides of a golden 2
rectangle is 5 "1 . In this lesson, you will learn to simplify radical expressions like 2
5!
"1 . Simplify Radicals: The properties you have used to simplify radical expressions involving square roots also hold true espressions involving nth roots. (page 439). n n2 n3 n4 n5 2 3 4 5 -­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐-­‐ In order for a radical to be insimplest form, the radical must contain no fators that are nth powers of an integer or polynomial. !
Directions: Simplify 4 9
a. 25a b 30 20
9 12 17
3
125m
p
"108x
y z b. c. 3
!
!is another property used to The quotient Property of Radicals simplify radicals. To eliminate radicals from a denominator or fractions from a radicand, you can use a process called rationalizing the denominator. To rationalize, multiply the numerator and denominator by quantity so that the radicand has an exact root. If the denominator is: Multiply the numerator & denominator by: n
b b b
x
n
b
Examples 2
=
5 n "x
! !
!
3
=
4
2 Directions: Simplify y8
7
a. x 2
b. 9x 3
8
c. 25 3
!
! of the rules used to simplify radicals: Here is a summary Operations with Radicals: You can use the Product and Quotient Properties to multiply and divide some radicals. Directions: Simplify 3
2
3
a. 5 100a • 10a !
3 2
5 2
4
4
2
8x
y
•
3
2x
y b. !
Radicals can be added and subtracted if the radicals are like terms. Radicals are like radical expressions if both the index and the radicand are identical. Directions: Simplify 3
4
3
a. 3 45 " 5 80 + 4 20 b. 2 125a " 5 8a You can also multiply radicals using the FOIL method as you do when multiplying b!
inomials. Directions: Simplify a. (2
3 + 3 5)(3 " 3) b. (7
2 " 3 3)(7 2 + 3 3) !
Binomials of the form a b + c d and a b " c d , where a, b, c, and d are rational numbers, are called conjugates of each other. You can use conjugates to rationalize denominators. !
!
ARCHITECTURE: Refer to the beginning of the lesson and simplify 2
the expression 5 "1 . !
6" 3
Simplify: 3 + 4 !
Notes 7.6-­‐ Rational Exponents Why? n
The formula C = c(1+ r) can be used to estimate the future cost of an item due to inflation. For example, 1
2
C!
= c(1+ r) can be used to estimate the cost of a video game system in six months. Rational Exponents and Radicals: Rational Exponents 1
n
Let b be an nth root of b, and let m be a positive integer. 1
n
n
• b = b m
n
!
•
b =
b
!
"m
n
=
( b)
n
1
m
n
m
1
=
m
,b # 0
( )
b
•
The rules for negative exponents also apply to negative rational !
exponents. n
b
!
1
7
7
4
a. Write a in radical form. b. Write a in radical form. orm. !c. Write w in exponential f!
d. Write ! 3
c "5 in exponential form. !
Directions: Evaluate each expression. "1
2
a. 49 !
2
5
b. 32 !
"2
3
c. 64 !
"3
5
d. ("243) e. !
3
10 "9 8
"
5
WEIGHTLIFTING: The formula M = 512 "146,230B can be used to estimate the maximum total mass that a weightlifter of mass B kilograms can lift using the snatch and the clean and jerk. According to the formula, what is the maximum that a weightlifter weighing 168 kilograms can lift? !
Simplify Expressions: All of the properties of powers you learned in Lesson 6-­‐1 apply to rational exponents. Write each expression with all positive exponents. Also, any exponents in the denominator of a fraction must be positive integers. So, it may be necessary to rationalize the denominator. Property Example Product of Power Property 1
4
3 •3
Power of a Power Property Power of a Product Property Negative Exponent Property 2 1
3 2
(5 )
!
!
Quotient of Powers Property 1
3 2
(64 • 8 )
"1
2
!
Power of a Quotient Property 1
3
36
3
4
7
!
7
3
4
1
4
2
!
!
"5 %
$ 2'
#8 &
1
2
Directions: Simplify each expression. 1
4
9
4
a. p • p b3
"2
3
b. x c. c
1
2
•
c
1
3
b !
!
Directions: Simplify each expression. 1
2
a. 3
16x
4
!
b. (27x 5 y
!
1
4 2
)
y +1
3
9 12 17
c. "64 x y z 1
2
d. y "1 !
CW/HW: Page 465# 41-­‐65 all and page 467 #19-­‐31 odd