Chapter 7

Square RootThe number b is a square root of a if b2 = a

Example 100 = 102 = 10

radical sign

Under radical sign the expression is called radicandExpression containing a radical sign is called a radical

expression.Radical expressions are 6, 5 + x + 1 , and 3x 2x - 1

Cube Root

The number b is a cube root of a if b3 = a

Example – Find the cube root of 27

3 27 = 3 33 = 3

Estimating a cellular phone transmission distance

R

The circular area A is covered by one transmission tower is A = R2 2

The total area covered by 10 towers are 10 R , which must equal to 50 square milesNow solve RR = 1.26, Each tower must broadcast with a minimum radius of approximately 1.26 miles

Expression For every real number If n is an integer greater than 1, then a1 n = n a

 Note : If a < 0 and n is an even positive integer, then

a1 n is not a real number.

If m and n are positive integer with m/n in lowest terms, then

a m n = n a m = ( n a ) m

 Note : If a < 0 and n is an even integer, then a m n

is not a real number. If m and n are positive integer with m/n in lowest terms, then

a - m n = 1/ a m n a = 0

Properties of ExponentLet p and q be rational numbers. For all real numbers a and b for which the expressions are real numbers the following properties hold.

a p . a q = a p + q Product rule

a - p = 1/ a p Negative exponents   a/b -p = b a p Negative exponents for quotients

a p = a p-q Quotient rule for exponents

a q

a p q = a pq Power rule for exponents ab p= a p b p Power rule for products

a p = a p Power rule for products

b b p Power rule for quotients

12

34

56

7

7.2 Simplifying Radical Expressions

Let a and b are real numbers where a and b are both defined.

Product rule for radical expression (Pg – 509)n a n b

=

n ab . =Quotient rule for radical expression where b = 0 (Pg 512)

=n

b

a n a

n b

n bn a ,

Square Root Property

Let k be a nonnegative number. Then the

solutions to the equation.

x2 = kare x = + k. If k < 0. Then this equation has no

real solutions.

Using Graphing Calculator

[ 5, 13, 1] by [0, 100, 10]

To find cube root technologically

7.3 Operations on Radical Expressions

Addition

10 + 4 = (10 + 4) = 14

Subtraction

11 11 11 11

10 - 4 = (10 - 4) = 611 11 11 11

3333 666)15(665

3333 646)15(665

Rationalize the denominator (Pg 484)

Using Graphing Calculator

Y1 = x2

[ -6, 6, 1] by [-4, 4, 1]

Pg -522

Rationalizing Denominators having square roots

3

1

3

3 = 3

3

7.6 Complex NumbersPg 556

x 2 + 1 = 0x 2 = -1x =+

- 1 Square root property

- 1i =

Now we define a number called the imaginary unit, denoted by i

Properties of the imaginary unit i

A complex number can be written in standard form, as a + bi, where a and b are real numbers. The real part is a and imaginary part is b

Pg 513

Complex Number -3 + 2i 5 -3i -1 + 7i - 5 – 2i 4 + 6i

Real part a - 3 5 -1 -5 4

Imaginary Part b 2 -3 7 -2 6

a + ib

Complex numbers contains the set of real numbers

Complex numbers

a +bi a and b real

Real numbersa +bi b=0

Imaginary Numbersa +bi b =0

Rational Numbers-3, 2/3, 0 and –1/2

Irrational numbers 3 And - 11

Sum or Difference of Complex Numbers

 Let a + bi and c + di be two complex numbers. Then

 Sum

( a + bi ) + (c +di) = (a + c) + (b + d)i

Difference

(a + bi) – (c + di) = (a - c) + (b – d)i