8.1 Radical Expressions and Graphs
means
is the principal nth root of a.
if nis even if nis odd.Functions Defined by Radical Expressions
The square root function defined by and the
cube root function defined by are two
important functions defined by radical expressions.
ƒ 1 x 2 23 xƒ 1 x 2 2 x2 nan=a
n
2 an=|a|n
2 abna.
n
2 ab The two square roots of 64 are (the principal square root)
and
241 - 224 =|- 2 |= 2 23 - 27 =- 3- 264 =-8.
264 = 8
QUICK REVIEW
CONCEPTS EXAMPLES
8.2 Rational Exponents
whenever exists.If mand nare positive integers with in lowest terms,
then provided that is a real number.
All of the usual definitions and rules for exponents are
valid for rational exponents.
am/n 1 a1/n 2 m, a1/nm
n2
n
a1/n a
n
2 a
(continued)484 CHAPTER 8 Roots, Radicals, and Root Functions
xy01 4 913f(x) = √xSquare root functionxy–8
8Cube root function12f(x) = √^3 x=
1
5 1/4
= 5 - 1/4
5 - 1/2# 5 1/4 = 5 - 1/2+1/4
8 5/3= 18 1/3 25 = 25 = 32 1 y2/5 210 =y^481 1/2= 281 = 9 - 64 1/3= - 2364 =- 4
=x1/6, x 70=x-1/3+1/2=x-1/3-^1 - 1/2^2x-1/3
x-1/28.3 Simplifying Radical Expressions
Product and Quotient Rules for Radicals
If and are real numbers and nis a natural
number, then
andConditions for a Simplified Radical
1.The radicand has no factor raised to a power greater
than or equal to the index.
2.The radicand has no fractions.
3.No denominator contains a radical.
4.Exponents in the radicand and the index of the radical
have greatest common factor 1.
Pythagorean Theorem
If aand bare the lengths of the shorter sides of a right
triangle and cis the length of the longest side, then
a^2 b^2 c^2.n bZ0.
Aa
bn
2 a
n
2 b,
2 na# 2 nb 2 nab
2 na 2 nb29 x^3 =x3/9=x1/3, or 23 xB
7
4
=
27
24
=
27
2
2354 x^5 y^3 = 2327 x^3 y^3 # 2 x^2 = 3 xy 232 x^2
218 = 29 # 2 = 322
2 x^5
2 x^4=
B
x^5
x^4= 2 x, x 7023 # 27 = 221 25 x^3 y# 25 xy^2 = 25 x^4 y^3
Find bfor the triangle in the figure.b= 12b^2 = 144b^2 = 41612 - 100102 +b^2 = A (^2261) B^2
10
b
2 √ 61
90