principal square root and cube root, algebra homework help

1. OPEN ENDED Write a number whose principal square root and cube root are both integers.
2. 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
• Use a calculator to
approximate radicals
GET READY for the Lesson
The radius r of a sphere with volume V can be
3V
found using the formula r = This is an
example of an equation that contains an nth root.
In this case, n = 3.
=
New Vocabulary
nth 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 n,
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
nth 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 5-4.
Powers
a? = 125
a4 = 81
a = 32
Factors
5.5.5 = 125
3.3.3.3 = 81
2.2.2.2.2 = 32
Roots
5 is a cube root of 125.
3 is a fourth root of 81.
2 is a fifth root of 32.
a = b
0.0.0.0…..q=b
n factors of a
a is an nth 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,
then a is an nth root of b.
Example Since 25 = 32, 2 is a fifth root of 32.
The symbol v indicates an nth root.
radical sign
index
V50
radicand
Some numbers have more than one real nth 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 Vb stands for the principal nth root of b. If n is odd and b is
negative, there will be no nonnegative root. In this case, the principal
root is negative.