Page 304 - Algebra 1

Higher order roots can be calculated in a similiar way that square roots are calculated. The cube root is a number, written as √3 x, whose cube is x. For example, the cube root of 8 is 2 because 23 = 8. The cube root of 8 is written as √3 8. In general, √n cn = c.
In the case of square roots, the index is not written and is understood to be 2.
Simplifying Roots
Simplify each expression.
a. √3 64 b. √3 - 8
SOLUTION SOLUTION
3√3 √3 6 4 = √3 4 √3 - 8 = 3 ( - 2 )
Higher−Order Roots
If an =b,thenthenthrootof bisa,or √n b =a.Thentotheleftof the radical sign in the expression is the index of the radical. The index is an integer greater than or equal to 2.
Example
2
=√3 4 · 4· 4=4
c. √4 81 SOLUTION
=√3 ( - 2) ·(- 2 )· (- 2)=-2
d. √4 - 1 SOLUTION
equivalent to the statement (√x)2 = x.
Simplifying Expressions with Fractional Exponents
Math Reasoning
Analyze Why does √4 - 1 have no real solution?
4 √4 8 1=√4 3
√4 - 1hasnorealsolution. Roots can be written as fractional exponents. Consider the expression
=√4 3 • 3• 3• 3=3
Using the Power of a Power Property, the expression simplifies to x. This is
( _1 ) 2 x 2 .
Fractional Exponents
n √n b = b _1
Example
3
Simplify each expression.
_1 a. (216) 3
SOLUTION
_1 b. (-27) 3
SOLUTION
Hint
In part c, notice that 10,000 = 1002, which equals (102)2 or 104. Sometimes it may be helpful to find the square root in order to find the nth root.
33
(216)_1 = √3 2 1 6 (-27)_1 = √3 - 2 7
= √3 6 = 6
= √3 ( - 3 ) = -3
33
(-256) _1 SOLUTION SOLUTION
44
(10,000)_1 =√4 1 0 ,00 0 (-256)_1 =√4 - 25 6hasnorealsolution.
4
=√4 10 =10
c. (10,000) _1
4 d.4
Lesson 46 289