6.2 Properties of Exponents
Lesson 6.2, For use with pages 420-427
Simplify the expression.
1. 43 • 48
ANSWER
411
85
2.
8–4
ANSWER
89
Lesson 6.2, For use with pages 420-427
Simplify the expression.
3. √48
ANSWER
4√3
4. √30 • √15
ANSWER
15√2
Lesson 6.2, For use with pages 420-427
Simplify the expression.
5. √80 – √20
ANSWER
2√5
EXAMPLE 1
Use properties of exponents
Use the properties of rational exponents to simplify the expression.
a. 71/4 71/2
= 73/4
= 7(1/4 + 1/2)
b. (61/2 41/3)2
= (61/2)2 (41/3)2
c. (45 35)–1/5
= [(4 3)5]–1/5
= 6(1/2 2) 4(1/3 2)
= (125)–1/5
= 12[5
(–1/5)]
= 61 42/3
= 12 –1
= 6 42/3
=
1
12
d.
5
=
51/3
e.
51
= 52/3
51/3
421/3 2
61/3
= 5(1 – 1/3)
=
42
6
1/3
2
= (71/3)2
= 7(1/3 2)
= 72/3
EXAMPLE 2
Apply properties of exponents
Biology
A mammal’s surface area S (in square centimeters) can be approximated
by the model S = km2/3 where m is the mass (in grams) of the mammal
and k is a constant. The values of k for some mammals are shown
below. Approximate the surface area of a rabbit that has a mass of 3.4
kilograms
(3.4 103 grams).
EXAMPLE 2
Apply properties of exponents
SOLUTION
S = km2/3
ANSWER
Write model.
= 9.75(3.4 103)2/3
Substitute 9.75 for k and 3.4 103
for m.
= 9.75(3.4)2/3(103)2/3
Power of a product property.
9.75(2.26)(102)
Power of a product property.
2200
Simplify.
The rabbit’s surface area is about 2200 square
centimeters.
GUIDED PRACTICE
for Examples 1 and 2
Use the properties of rational exponents to simplify the expression.
1. (51/3 71/4)3
SOLUTION
5 73/4
2. 23/4 21/2
SOLUTION
25/4
SOLUTION
33/4
SOLUTION
8
3.
3
31/4
4.
201/2
51/2
3
GUIDED PRACTICE
for Examples 1 and 2
Biology
5.
Use the information in Example 2 to approximate the surface area of
a sheep that has a mass of 95 kilograms (9.5 104 grams).
SOLUTION
17,500 cm2
EXAMPLE 3
Use properties of radicals
Use the properties of radicals to simplify the expression.
3
a.
b.
4
4
12
80
5
3
18
=
=
4
3
80
5
12 8
=
4
=
16
3
=
216 = 6
2
Product property
Quotient property
EXAMPLE 4
Write radicals in simplest form
Write the expression in simplest form.
a.
3
 135
3
=
 27
5
=
 27
3
3
=
3
3 5
 5
Factor out perfect cube.
Product property
Simplify.
EXAMPLE 4
Write radicals in simplest form
5
b.
 7
5
 8
=
5
 7
5
5
5
 8
 4
 4
Make denominator a perfect fifth
power.
5
 28
=
5
 32
Product property
5
 28
=
2
Simplify.
EXAMPLE 5
Add and subtract like radicals and roots
Simplify the expression.
4
4
+ 7  10 = (1 + 7)
4
 10
b.
2 (81/5) + 10 (81/5) = (2 +10)
c.
3
3
3
 54 –  2 =  27
4
 10 = 8  10
a.
(81/5) = 12 (81/5)
3
3
3
 2 –  2 = 3  2 –  2 = (3 – 1)
3
3
 2 = 2 3 2
for Examples 3, 4, and 5
GUIDED PRACTICE
Simplify the expression.
6.
4
27
4
3
8.
5
3
4
5
SOLUTION
3
 24
SOLUTION
2
7.
3
 250
9.
3
 5
3
+  40
3
 2
SOLUTION
5
SOLUTION
3
3 5
EXAMPLE 6
Simplify expressions involving variables
Simplify the expression. Assume all variables are positive.
3
a.
3
b.
(27p3q12)1/3
4
c.
d.
m4
n8
3
=  43(y2)3
 64y6
14xy 1/3
2x 3/4 z –6
 (y2)3
= 271/3(p3)1/3(q12)1/3
4
=
3
=  43

4
m4
 n8
4
=
 m4
4
 (n2)4
= 7x(1 – 3/4)y1/3z –(–6)
=
= 3p(3
= 4y2
1/3)q(12 1/3)
m
n2
= 7x1/4y1/3z6
= 3pq4
EXAMPLE 7
Write variable expressions in simplest form
Write the expression in simplest form. Assume all variables are positive.
a.
5
5
 4a8b14c5
=  4a5a3b10b4c5
5
=  a5b10c5
b.
3
x
y8
3
=
=
3
5
 4a3b4
5
=
ab2c  4a3b4
x
y
y8
y
Product property
Simplify.
Make denominator a perfect cube.
xy
y9
Factor out perfect fifth
powers.
Simplify.
EXAMPLE 7
Write variable expressions in simplest form
3
=
 xy
3

Quotient property
y9
3
=
 xy
y3
Simplify.
EXAMPLE 8
Add and subtract expressions involving variables
Perform the indicated operation. Assume all variables are positive.
a.
b.
c.
1
3
w +
w =
5
5
3xy1/4 – 8xy1/4
3
3
1
+
5
5
w
= (3 – 8) xy1/4
3

12 2z5 – z  54z2
=
4
w
5
= –5xy1/4
3
3
= 12z  2z2 – 3z  2z2
3
= (12z – 3z)  2z2
3
= 9z  2z2
for Examples 6, 7, and 8
GUIDED PRACTICE
Simplify the expression. Assume all variables are positive.
10.
3
 27q9
SOLUTION
11.
5
12.
3q3
x10
3x 1/2 y 1/2
SOLUTION
13.
y5
SOLUTION
6xy 3/4
x2
y
2x1/2y1/4
 9w5 – w  w3
SOLUTION
2w2  w