MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C923D0.455D6C50" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01C923D0.455D6C50 Content-Location: file:///C:/5F241E43/M117Lesson10.1doc.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" To simplify numerical expressions

Lesson 10.1(F’08)

 

 

Objectives:      To find the= nth root of a number.

        &= nbsp;           &nbs= p;   To approximate roots using a calculator.

        &= nbsp;           &nbs= p;   To simplify radical expressions.

        &= nbsp;           &nbs= p;   To evaluate radical functions.

To solve applications involving radical function= s.

 

22 =3D 2  2 =3D 4            = ;            &n= bsp;     (-3)2 =3D (-3)(-3) =3D 9

 

The square root of a number is ONE<= /span> of its two equal fact= ors.

 

Find the square root of:

1.             &= nbsp;           &nbs= p;            &= nbsp;   or is it -8?

 

2. 

 

GENERALIZATIONS:

1.      Every positive real nu= mber has two square roots: one is positive and one is negative. 

     They are oppo= sites of each other.

 

2.      Negative real numbers = have no real number square roots because any nonzero real number is positive when squared.   does not ex= ist in the real numbers.

 

3.      The square root of 0 i= s 0.

 

3D"*"    &= nbsp;        radical sign

radicand is 64

  &nbs= p;            &= nbsp;    the radical is

If  and , then  =3D a if an= d only if a2 =3D b:

 a= is called the pr= incipal square root of b. 

 

When you are asked to= find , you must give the positive answer (principal square roo= t)

 

Evaluate the followin= g roots.

 

1. 

 

 

2. 

 

 

3. 

 

 

4. 

 

Cube roots

 

23 =3D 2 22 =3D 8       (-3)3 – (-3)(-3)(-3) =3D -27

 

The cube root of a number is ONE o= f its three equal factors.

            =             &nb= sp;   3D"*"=3D a if and only if a3 =3D b.

Evaluate the following roots.

 

 

GENERALIZATIONS:=

1.&n= bsp;     Every posit= ive real number has one positive r= eal number cube root.

 

2.&n= bsp;     Every negat= ive number has one negative real n= umber cube root.

 

3.&n= bsp;     The cube ro= ot of 0 is 0.

 

nth roots

            =             &nb= sp;   3D"*"=3D a if and only if an =3D b.

 

If n is an even positive integer, then the following statements are true:<= /o:p>

1.      Every positive real nu= mber has exactly two real nth

roots--one positive and one negative.  For example,

  &nb= sp; the real fourth roots of 16 are 2 and –2.

2.      Negative real numbers = do not have real nth roots.  For

  &nb= sp; example, there are no real fourth roots of –16.

If n is an odd positive integer g= reater than 1, then the following statements are true:

1.      Every real number has exactly one real nth root.

2.      The real nth root of a positive number is positive.  = For example, the fifth root of 32 is 2.

3.      The real nth root of a negative number is negative.  = For example, the fifth root of –32 is –2.

 

Let’s use our calculator to approximate some roots.  Round to three decimals.

 

1. 

 

2. 

 

3. 

 

Now let’s find = the roots of terms with a variable in the radicand.

 

Find each of the foll= owing roots.  Assume that all variab= les represent nonnegative values.

 

1.      &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p; 2. 

 

3.      &= nbsp;           &nbs= p;            &= nbsp;       4. 

 

5.        

If the variables can represent any real number and the index is even, we must = make sure that the answer is nonnegative.  Remember we must always give the principal root.  We will do this by using absolute = value signs.

 

Find the roots.  Assume that variab= les represent any real number.

 

1.          &= nbsp;           &nbs= p;            &= nbsp;   2. 

 

 

 

3.          &= nbsp;           &nbs= p;        4. 

 

 

 

5. 

 

 

 

RADICAL FUNCTION:  A function of the f= orm f(x) =3D ,

   &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;        w= here P is a polynomial.

 

GIVEN            f(x) =3D   Find:

 

1.  f(0)            &= nbsp;        2.  f(6)            &= nbsp;        3. f(-1)

 

 

 

 

Let’s look at t= he graph of f(x) =3D , we can see that the do= main is NOT all real numbers.  What= is it?

3D"[image]"

 

If we graph f(x) =3D = , we can see that the do= main is all real number.

3D"[image]"

 

 

Find the domain of ea= ch of the following.

 

1.  f(x) =3D             =           2.  f(x) =3D 

 

 

 

 

 

 

 

Now let’s look = at the graph of each and see if our domain is correct.

3D"[image]"3D"[image]"

 

Many real world appli= cations involve radical expressions.  = For example, police often use the formula S =3D

to estimate the speed= of a car on the basis of the length of skid marks.  In this formula, S represents the = speed of the car in miles per hour, D represents the length of skid marks measure= d in feet, and f represents a coefficient of friction.  For a particular situation, the coefficient of friction is a constant that depends on the type and conditio= n of the road.=

1.      Use 0.25 as a coeffici= ent of friction, determine how fast a car was traveling if it skidded 225 feet.

 

 

 

 

 

 

 

 

 

 

Radical expressions a= re also used in some geometric applications.  For example, if a, b, and c represent the lengths of the three sides= of a triangle, the formula

K =3D  known as heron’s formula, can be used to determine the area (K) of the triangle.  The letter s repres= ents the semi-perimeter of the triangle; that is

s =3D  .

2.      Find the area of a triangular piece of cardboard that has sides of lengths 16in, 18in, and 14in. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CW-Worksheet 10.1

Homework Math XL Section 10.1

 

DUE BY ____________________

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