explain how the nth roots apply to geometry, algebra homework help

Writing in Math Refer to the information on

page 422

to explain how the nth roots apply to geometry. Analyze what happens to the value of r as the value of V increases.

7-4
nth Roots
Main Ideas
Simplify radicals
GET READY for the Lesson
The radius r of a sphere with volume V can be
found using the formular = This is an
example of an equation that contains an nth root.
In this case, n = 3.
• Use a calculator to
approximate radicals
New Vocabulary
nith root
principal root
Simplify Radicals Finding the square root of a number and squaring a
number are inverse operations. To find the square root of a number 1,
you must find a number whose square is n.
Study Tip
Similarly, the inverse of raising a number to the nth power is finding the
wth root of a number. The table below shows the relationship between
raising a number to a power and taking that root of a number
Look Back
Review square roots
in Lesson 54.
Powers
Roots
Factors
5.5.5 = 125
a = 125
d’ = 81
= 32
3 – 3 – 3 – 3 = 81
5 is a cube root of 125
3 is a fourth root of 81.
2 is a fifth root of 32
2.2.2.2.2 = 32
0.0.0.0…..
n factors of a
a = b
a is an ath root of b.
This pattern suggests the following formal definition of an nth root
KEY CONCEPT
Definition of nth Root
Word For any real numbers a and b, and any positive integer n, if a = b.
thena is an nth root of b.
Example Since 29 = 32, 2 is a fifth root of 32.
The symbol V indicates an nth root.
radical sign
index
radicand
Some numbers have more than one real th root. For example, 36 has
two square roots, 6 and – 6. When there is more than one real root, the
nonnegative root is called the principal root. When no index is given, as
in 36, the radical sign indicates the principal square root. The
symbol V stands for the principal nith root of b. If is odd and bis
negative, there will be no nonnegative root. In this case, the principal
root is negative.
402 Chapter 7 Radical Fouations and inequalities