Radical Expressions, Equations, and CHAPTER · PDF fileEvery positive real number has two real-number square roots. The number 0 has just one square root, 0 itself

CHAPTER

10 Radical Expressions, Equations, and Functions

Slide 2 Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

10.1 Radical Expressions and Functions 10.2 Rational Numbers as Exponents 10.3 Simplifying Radical Expressions 10.4 Addition, Subtraction, and More Multiplication 10.5 More on Division of Radical Expressions 10.6 Solving Radical Equations 10.7 Applications Involving Powers and Roots 10.8 The Complex Numbers

OBJECTIVES

10.1 Radical Expressions and Functions


a Find principal square roots and their opposites, approximate square roots, identify radicands, find outputs of square-root functions, graph square-root functions, and find the domains of square-root functions.

b Simplify radical expressions with perfect-square radicands.

c Find cube roots, simplifying certain expressions, and find outputs of cube-root functions.

OBJECTIVES



d Simplify expressions involving odd roots and even roots.


Square Root


The number c is a square root of a if c2 = a.


Properties of Square Roots


Every positive real number has two real-number square roots. The number 0 has just one square root, 0 itself. Negative numbers do not have real-number square roots.

EXAMPLE


a Find principal square roots.

1 Find the square root.


Find the two square roots of 64.

The square roots of 64 are 8 and –8 because 82 = 64 and (–8)2 = 64.


Principle Square Root


The principal square root of a nonnegative number is its nonnegative square root. The symbol represents the principal square root of a. To name the negative square root of a, write

EXAMPLE



Simplify.


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Does not exist as a real number. Negative numbers do not have real-number square roots.

EXAMPLE


a Find approximate square roots.

Approximate.


EXAMPLE Solution


a Find approximate square roots.



Radical; Radical Expression; Radicand


The symbol is called a radical. An expression written with a radical is called a radical expression. The expression written under the radical is called the radicand.

EXAMPLE


a Find outputs of square-root functions

14 Find the indicated function values.


For the given function, find the indicated function values:

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a Find outputs of square-root functions

14


EXAMPLE


a Find the domains of square-root functions.

15 Find the domain.


Find the domain of

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a Find the domains of square-root functions.

15


The expression is a real number only when x + 2 is nonnegative. Thus the domain of is the set of all x-values for which

The domain of

EXAMPLE


a Graph square-root functions.

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Graph.

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First find outputs. Once ordered pairs have been calculated, a smooth curve can be drawn.

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See from the table and the graph that the domain of f is The range is also the set of nonnegative real numbers



16


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The domain of is The range is the set of nonnegative real numbers


Simplifying


If a is positive or 0, the principal square root of a2 is a. If a is negative, the principal square root of a2 is the opposite of a.

For any real number The principal (nonnegative) square root of a2 is the absolute value of a.


Principle Square Root of a2


EXAMPLE



Simplify.


Find each of the following. Assume that letters can represent any real number.

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Cube Root




Every real number has exactly one cube root in the system of real numbers. The symbol represents the cube root of a.

EXAMPLE



Find.


EXAMPLE Solution




EXAMPLE



29 Find the function values.


For the given function, find the indicated function values:

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29







If k is an odd natural number, then for any real number a,

EXAMPLE



Find the following.


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When the index k in is an even number, we say that we are taking an even root. When the index is 2, do not write it. Every positive real number has two real-number kth roots when k is even. One of those roots is positive and one is negative. Negative real numbers do not have real-number kth roots when k is even.

EXAMPLE



Find the following.


Assume that variables can represent any real number.

EXAMPLE Solution




For any real number : a) when k is an even natural number. Use

absolute value when k is even unless a is nonnegative.

b) when k is an odd natural number greater

than 1. Do not use absolute value when k is odd.


Simplifying


CHAPTER




OBJECTIVES

10.2 Rational Numbers as Exponents


a Write expressions with or without rational exponents, and simplify, if possible.

b Write expressions without negative exponents, and simplify, if possible.

c Use the laws of exponents with rational exponents. d Use rational exponents to simplify radical expressions.

For any nonnegative real number a and any natural number index n a1/n means (the nonnegative nth root of a).


a1/n


EXAMPLE



Rewrite without rational exponents..


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EXAMPLE



Rewrite with rational exponents.


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am/n


For any natural numbers m and n and any nonnegative real number a,

EXAMPLE



7 Rewrite without rational exponents and simplify.


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7


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10 Rewrite with rational exponents.


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The index becomes the denominator of the rational exponent.



For any rational number m/n and any positive real number a, that is, am/n and a–m/n are reciprocals.

EXAMPLE



Rewrite with positive exponents and simplify.


EXAMPLE Solution






For any real number a and any rational exponents m and n: 1. In multiplying, we can add

exponents if the bases are the same. 2. In dividing, we can subtract

exponents if the bases are the same. 3. To raise a power to a power, we

can multiply the exponents.



4. To raise a product to a power, we

can raise each factor to the power. 5. To raise a quotient to a power,

we can raise both the numerator and the denominator to the power.

EXAMPLE


c Use the laws of exponents with rational exponents.

Simplify.


EXAMPLE Solution


c Use the laws of exponents with rational exponents.



Simplifying Radical Expressions


1. Convert radical expressions to exponential expressions.

2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation when appropriate. Important: This procedure works only when we assume that a negative number has not been raised to an even power in the radicand. With this assumption, no absolute-value signs will be needed.

EXAMPLE


d Use rational exponents to simplify radical expressions.

Use rational exponents to simplify.


EXAMPLE Solution




EXAMPLE



23 Use rational exponents to write a single radical expression.


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24 Write a single radical expression.


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25 Write a single radical expression.


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EXAMPLE



Use rational exponents to simplify.


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CHAPTER




OBJECTIVES

10.3 Simplifying Radical Expressions


a Multiply and simplify radical expressions. b Divide and simplify radical expressions.


The Product Rule for Radicals


For any nonnegative real numbers a and b and any index k,

(To multiply, multiply the radicands.)

EXAMPLE


a Multiply and simplify radical expressions.

Multiply.


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EXAMPLE



5 Multiply.


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5


For any nonnegative real numbers a and b and any index k, (Take the kth root of each factor separately.)


Factoring Radical Expressions



Simplifying kth Roots


To simplify a radical expression by factoring: 1. Look for the largest factors of the radicand that are

perfect kth powers (where k is the index). 2. Then take the kth root of the resulting factors. 3. A radical expression, with index k, is simplified when

its radicand has no factors that are perfect kth powers.

EXAMPLE



Simplify.


EXAMPLE Solution




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EXAMPLE



9 Simplify by factoring.


EXAMPLE



Simplify by factoring.


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EXAMPLE



Multiply and simplify.


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The Quotient Rule for Radicals


For any nonnegative number a, any positive number b, and any index k, (To divide, divide the radicands. After doing this, you can sometimes simplify by taking roots.)

EXAMPLE


b Divide and simplify radical expressions.

Divide and simplify.


Assume that no radicands were formed by raising negative numbers to even powers.

EXAMPLE Solution





kth Roots of Quotients


For any nonnegative number a, any positive number b, and any index k, (Take the kth roots of the numerator and of the denominator separately.)

EXAMPLE



Simplify.


Simplify by taking the roots of the numerator and the denominator. Assume that no radicands were formed by raising negative numbers to even powers.

EXAMPLE Solution




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EXAMPLE



23 Divide and simplify.


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23


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CHAPTER




OBJECTIVES

10.4 Addition, Subtraction, and More Multiplication


a Add or subtract with radical notation and simplify. b Multiply expressions involving radicals in which

some factors contain more than one term.


a Add or subtract with radical notation and simplify.


Like radicals are radicals having the same index and radicand.

EXAMPLE



Add or subtract.


Simplify by collecting like radical terms, if possible.

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EXAMPLE



4 Add or subtract.



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4


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6 Add or subtract.



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EXAMPLE


b Multiply expressions involving radicals in which some factors contain more than one term.

8 Multiply.


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8


EXAMPLE



9 Multiply.


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EXAMPLE



11 Multiply.


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EXAMPLE



13 Multiply.


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CHAPTER




OBJECTIVES

10.5 More on Division of Radical Expressions


a Rationalize the denominator of a radical expression having one term in the denominator.

b Rationalize the denominator of a radical expression having two terms in the denominator.




An equivalent expression without a radical in the denominator provides a standard notation for expressing results. The procedure for finding such an expression is called rationalizing the denominator. Carry this out by multiplying by 1.

EXAMPLE



1 Rationalize the denominator.


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1


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Assume that no radicands were formed by raising negative numbers to even powers.

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are called conjugates. The product of such a pair of conjugates has no radicals in it. Thus when we wish to rationalize a denominator that has two terms and one or more of them involves a square-root radical, multiply by 1 using the conjugate of the denominator to write a symbol for 1.




EXAMPLE





EXAMPLE





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CHAPTER




OBJECTIVES

10.6 Solving Radical Equations


a Solve radical equations with one radical term. b Solve radical equations with two radical terms. c Solve applied problems involving radical equations.


a Solve radical equations with one radical term.


A radical equation has variables in one or more radicands.


The Principle of Powers


For any natural number n, if an equation a = b is true, then an = bn is true.

EXAMPLE



1 Solve.


The number 49 is a possible solution. But we must check in order to be sure! The solution is 49.

EXAMPLE



3 Solve.


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3


The radical term is already isolated. Proceed with the principle of powers:

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3


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3


Check.

The solution is 15.

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5 Solve.


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Solving Radical Equations


To solve radical equations: 1. Isolate one of the radical terms. 2. Use the principle of powers. 3. If a radical remains, perform steps (1) and (2) again. 4. Check possible solutions.

EXAMPLE


b Solve radical equations with two radical terms.

6 Solve.


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6


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6


EXAMPLE



7 Solve.


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7


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7


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Check.

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7


The numbers 7 and 3 check and are the solutions.

EXAMPLE


c Solve applied problems involving radical equations.

9 Outdoor Concert.


The geologically formed, open-air Red Rocks Amphitheatre near Denver, Colorado, hosts a series of concerts. A scientific instrument at one of these concerts determined that the sound of the music was traveling at a rate of 1170 ft/sec. What was the air temperature at the concert?

EXAMPLE Solution



9


Substitute 1170 for S in the formula

Then solve the equation for t:

EXAMPLE Solution



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CHAPTER




OBJECTIVES

10.7 Applications Involving Powers and Roots


a Solve applied problems involving the Pythagorean theorem and powers and roots.

EXAMPLE



2 Find the length of the hypotenuse.


Find the length of the hypotenuse of this right triangle. Give an exact answer and an approximation to three decimal places.

EXAMPLE Solution



2


EXAMPLE



3 Find the missing length.


Find the missing length in this right triangle. Give an exact answer and an approximation to three decimal places.

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3


EXAMPLE

In a psychological study, it was determined that the ideal length of the letters of a word painted on pavement is given by where d is the distance of a car from the lettering and h is the height of the eye above the road. All units are in feet.



5 Road-Pavement Messages.


EXAMPLE



5 Road-Pavement Messages.


For a person h feet above the road, a message d feet away will be the most readable if the length of the letters is L. Find L, given that h = 4 ft and d = 180 ft.

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5


Substitute 4 for h and 180 for d and calculate L using a calculator.

CHAPTER




OBJECTIVES

10.8 The Complex Numbers


a Express imaginary numbers as bi, where b is a nonzero real number, and complex numbers as a + bi where a and b are real numbers.

b Add and subtract complex numbers. c Multiply complex numbers. d Write expressions involving powers of i in the form

a + bi. e Find conjugates of complex numbers and divide

complex numbers.

OBJECTIVES



f Determine whether a given complex number is a solution of an equation.


a Complex numbers.


The complex-number system is a larger number system that contains the real-number system, such that negative numbers have square roots.


The Complex Number i


EXAMPLE


a Express complex numbers as a + bi where a and b are real numbers.

Express in terms of i.


EXAMPLE Solution


a Express complex numbers as a + bi where a and b are real numbers.



Imaginary Number


An imaginary number is a number that can be named bi, where b is some real number and b ≠ 0.


Complex Number


A complex number is any number that can be named a + bi, where a and b are any real numbers. (Note that either a or b or both can be 0.)


a Express imaginary numbers and complex numbers.











EXAMPLE


b Add and subtract complex numbers.

Add or subtract.


EXAMPLE Solution


b Add and subtract complex numbers.


EXAMPLE


c Multiply complex numbers.

Multiply.


EXAMPLE Solution




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d Write expressions involving powers of i in the form a + bi.


The powers of i cycle through the values i, –1, –i, and 1.

EXAMPLE



Simplify.


EXAMPLE Solution




EXAMPLE



Simplify to the form a + bi.


EXAMPLE Solution





Conjugate


The conjugate of a complex number a + bi is a – bi and the conjugate of a – bi is a + bi.

EXAMPLE


e Find conjugates of complex numbers and divide complex numbers.

Find the conjugate.


EXAMPLE Solution






When we multiply a complex number by its conjugate, we get a real number.

EXAMPLE



27 Multiply.


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27


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30 Divide.


Divide and simplify to the form a + bi:

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30


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31 Determine the solution.


Determine whether i is a solution of the equation x2 + 1 = 0.

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31


Substitute i for x in the equation.

The number i is a solution.

EXAMPLE



33 Determine the solution.


Determine whether 2i is a solution of x2 + 3x – 4 = 0.

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Radical Expressions, Equations, and CHAPTER · PDF fileEvery positive real number has two real-number square roots. The number 0 has just one square root, 0 itself

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