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Square Roots

- A square root of a real number “a” is a real number that multiplies by itself to give “a”

What is a square root of 9?

What is another square root of 9?

- What is the square root of -4 ?

Square root of – 4 does not exist in the real number system

- Why is it that square roots of negative numbers do not exist in the real number system?

No real number multiplied by itself can give a negative answer

- Every positive real number “a” has two square roots that have equal absolute values, but opposite signs

The two square roots of 16 are:

The two square roots of 5 are:

Even Roots (2,4,6,…)

- The even “nth” root of a real number “a” is a real number that multiplies by itself “n” times to give “a”
- Even roots of negative numbers do not existin the real number system, because no real number multiplied by itself an even number of times can give a negative number
- Every positive real number “a” has two even roots that have equal absolute values, but opposite signs

The fourth roots of 16:

The fourth roots of 7:

Radical Expressions

- On the previous slides we have used symbols of the form:
- This is called a radical expression and the parts of the expression are named:

Index:

Radical Sign :

Radicand:

- Example:

Cube Roots

- The cube root of a real number “a” is a real number that multiplies by itself 3 times to give “a”
- Every real number “a” has exactly one cube root that is positive when “a” is positive, and negative when “a” is negative

Only cube root of – 8:

Only cube root of 6:

Odd Roots (3,5,7,…)

- The odd nth root of a real number “a” is a real number that multiplies by itself “n” times to give “a”
- Every real number “a” has exactly one odd root that is positive when “a” is positive, and negative when “a” is negative

The only fifth root of - 32:

The only fifth root of -7:

Rational, Irrational, and Non-real Radical Expressions

- is non-real only if the radicand is negative and the index is even
- represents a rational number only if the radicand can be written as a “perfect nth” power of an integer or the ratio of two integers
- represents an irrational number only if it is a real number and the radicand can not be written as “perfect nth” power of an integer or the ratio of two integers

.

Homework Problems

- Section: 10.1
- Page: 666
- Problems: All: 1 – 6, Odd: 7 – 31, 39 – 57, 65 – 91
- MyMathLab Homework Assignment 10.1 for practice
- MyMathLab Quiz 10.1 for grade

Exponential Expressions

an

“a” is called the base

“n” is called the exponent

- If “n” is a natural number then “an” means that “a” is to be multiplied by itself “n” times.

Example: What is the value of 24 ?

(2)(2)(2)(2) = 16

- An exponent applies only to the base (what it touches)

Example: What is the value of: - 34 ?

- (3)(3)(3)(3) = - 81

Example: What is the value of: (- 3)4 ?

(- 3)(- 3)(- 3)(- 3) = 81

- Meanings of exponents that are not natural numbers will be discussed in this unit.

Negative Exponents: a-n

- A negative exponent has the meaning: “reciprocate the base and make the exponent positive”

Examples:

.

Quotient Rule for Exponential Expressions

- When exponential expressions with the samebase are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent

Examples:

.

Rational Exponents (a1/n)and Roots

- An exponent of the form

has the meaning: “the nth root of the base, if it exists, and, if there are two nth roots, it means the principle (positive) one”

Summary Comments about Meaning of a1/n

- When n is odd:
- a1/n always exists and is either positive, negative or zero depending on whether “a” is positive, negative or zero
- When n is even:
- a1/n never exists when “a” is negative
- a1/n always exists and is positive or zero depending on whether “a” is positive or zero

Rational Exponents of the Form: m/n

- An exponent of the form m/n has two equivalent meanings:

(1) am/n means find the nth root of “a”, then raise it to the power of “m”

(assuming that the nth root of “a” exists)

(2) am/n means raise “a” to the power of “m” then take the nth root of am

(assuming that the nth root of “am” exists)

Example of Rational Exponent of the Form: m/n

82/3

by definition number 1 this means find the cube root of 8, then square it:

82/3 = 4

(cube root of 8 is 2, and 2 squared is 4)

by definition number 2 this means raise 8 to the power of 2 and then cube root that answer:

82/3 = 4

(8 squared is 64, and the cube root of 64 is 4)

Definitions and Rules for Exponents

- All the rules learned for natural number exponents continue to be true for both positive and negative rational exponents:

Product Rule: aman = am+n

Quotient Rule: am/an = am-n

Negative Exponents: a-n = (1/a)n

.

Definitions and Rules for Exponents

Power Rules: (am)n = amn

(ab)m = ambm

(a/b)m = am / bm

Zero Exponent: a0 = 1 (a not zero)

.

“Slide Rule” for Exponential Expressions

- Whenboth the numeratoranddenominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponent

Example: Use rule to slide all factors to other part of the fraction:

- This rule applies to all types of exponents
- Often used to make all exponents positive

Simplifying Products and Quotients Having Factors withRational Exponents

- All factors containing a common base can be combined using rules of exponents in such a way that all exponents are positive:
- Use rules of exponents to get rid of parentheses
- Simplify top and bottom separately by using product rules
- Use slide rule to move all factors containing a common base to the same part of the fraction
- If any exponents are negative make a final application of the slide rule

Applying Rules of Exponentsin Multiplying and Factoring

- Multiply:
- Factor out the indicated factor:

Radical Notation

- Roots of real numbers may be indicated by means of either rational exponent notation or radical notation:

Notes About Radical Notation

- If no index is shown it is assumed to be 2
- When index is 2, the radical is called a “square root”
- When index is 3, the radical is called a “cube root”
- When index is n, the radical is called an “nth root”
- In the real number system, we can only find even roots of non-negative radicands. There are alwaystwo roots when the index is even, but a radicalwith an even index always means the positive (principle) root
- We can always find an odd root of any real number and the result is positive or negative depending on whether the radicand is positive or negative

Converting Between Radical and Rational Exponent Notation

- An exponential expression with exponent of the form “m/n” can be converted to radical notation with index of “n”, and vice versa, by either of the following formulas:

1.

2.

- These definitions assume that the nth root of “a” exists

Examples

.

.

- If “n” is even, then this notation means principle (positive) root:
- If “n” is odd, then:
- If we assume that “x” is positive (which we often do) then we can say that:

.

Homework Problems

- Section: 10.2
- Page: 675
- Problems: All: 1 – 10, Odd: 11 – 47, 51 – 97
- MyMathLab Homework Assignment 10.2 for practice
- MyMathLab Quiz 10.2 for grade

Product Rule for Radicals

- When two radicals are multiplied that have the same index they may be combined as a single radical having that index and radicand equal to the product of the two radicands:
- This rule works both directions:

Quotient Rule for Radicals

- When two radicals are divided that have the same index they may be combined as a single radical having that index and radicand equal to the quotient of the two radicands
- This rule works both directions:

.

Root of a Root Rule for Radicals

- When you take the mth root of the nth root of a radicand “a”, it is the same as taking a single root of “a” using an index of “mn”

.

Simplifying Radicals

- A radical must besimplified if any of the following conditions exist:
- Some factor of the radicand has an exponent that is bigger than or equal to the index
- There is a radical in a denominator (denominator needs to be “rationalized”)
- The radicand is a fraction
- All of the factors of the radicand have exponents that share a common factor with the index

Simplifying when Radicand has Exponent Too Big

- Use the product rule to write the single radicalas a product of two radicals where the first radicand contains allfactorswhose exponents match the index and the second radicand contains all other factors
- Simplify the first radical

Simplifying when a Denominator Contains a Single Radical of Index “n”

- Simplify the top and bottom separately to get rid of exponents under the radical that are too big
- Multiply the whole fraction by a special kind of “1” where 1 is in the form of:
- Simplify to eliminate the radical in the denominator

Simplifying when Radicand is a Fraction

- Use the quotient rule to write the singleradicalas a quotient of two radicals
- Use the rules already learned for simplifying when there is a radical in a denominator

Simplifying when All Exponents in Radicand Share a Common Factor with Index

- Divide out the common factor from the index and all exponents

Simplifying Expressions Involving Products and/or Quotients of Radicals with the Same Index

- Use the product and quotient rules to combine everything under a singleradical
- Simplify the single radical by procedures previously discussed

Right Triangle

- A “right triangle” is a triangle that has a 900 angle (where two sides intersect perpendicularly)
- The side opposite the right angle is called the “hypotenuse” and is traditionally identified as side “c”
- The other two sides are called “legs” and are traditionally labeled “a” and “b”

Pythagorean Theorem

- In a right triangle, the square of the hypotenuse is always equal to the sum of the squares of the legs:

Pythagorean Theorem Example

- It is a known fact that a triangle having shorter sides of lengths 3 and 4, and a longer side of length 5, is a right triangle with hypotenuse 5.
- Note that Pythagorean Theorem is true:

Using the Pythagorean Theorem

- We can use the Pythagorean Theorem to find the third side of a right triangle, when the other two sides are known, by finding, or estimating, the square root of a number

Using the Pythagorean Theorem

- Given two sides of a right triangle with one side unknown:
- Plug two known values and one unknown value into Pythagorean Theorem
- Use addition or subtraction to isolate the “variable squared”
- Square root both sides to find the desired answer

Example

- Given a right triangle with find the other side.

Homework Problems

- Section: 10.3
- Page: 685
- Problems: Odd: 7 – 19, 23 – 57, 61 – 107
- MyMathLab Homework Assignment 10.3 for practice
- MyMathLab Quiz 10.3 for grade

Adding and Subtracting Radicals

- Addition and subtraction of radicals can always be indicated, but can be simplifiedinto a single radicalonlywhen the radicals are “like radicals”
- “Like Radicals” are radicals that have exactly the same index and radicand, but may have different coefficients

Which are like radicals?

- When “like radicals” are added or subtracted, the result is a “like radical” with coefficient equal to the sum or difference of the coefficients

Note Concerning Adding and Subtracting Radicals

- When addition or subtraction of radicals is indicated you must first simplify all radicals because some radicals that do not appear to be like radicals become like radicals when simplified

Homework Problems

- Section: 10.4
- Page: 691
- Problems: Odd: 5 – 57
- MyMathLab Homework Assignment 10.4 for practice
- MyMathLab Quiz 10.4 for grade

Simplifying when there is a Single Radical Term in a Denominator

- Simplify the radical in the denominator
- If the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special radical” over itself
- The “special radical” is one that contains the factors necessary to make the denominator radical factors have exponents equal to index
- Simplify radical in denominator to eliminate it

Simplifying to Get Rid of a Binomial Denominator that ContainsOne orTwoSquare Root Radicals

- Simplify the radical(s) in the denominator
- If the denominator still contains a radical, multiply the fraction by “1” where “1” is in the form of a “special binomial radical” over itself
- The “special binomial radical” is the conjugate of the denominator (same terms – opposite sign)
- Complete multiplication (the denominator will contain no radical)

Homework Problems

- Section: 10.5
- Page: 700
- Problems: Odd: 7 – 105
- MyMathLab Homework Assignment 10.5 for practice
- MyMathLab Quiz 10.5 for grade

Radical Equations

- An equation is called a radical equation if it contains a variable in a radicand
- Examples:

Solving Radical Equations

- Isolate ONE radical on one side of the equal sign
- Raise both sides of equation topower necessary to eliminate the isolated radical
- Solve the resulting equation to find “apparent solutions”
- Apparent solutions will be actual solutionsif both sides of equation were raised to an odd power, BUTif both sides of equation were raised to an even power, apparent solutionsMUSTbechecked to see if they are actual solutions

Why Check When Both Sides are Raised to an Even Power?

- Raising both sides of an equation to a power does not always result in equivalent equations
- If both sides of equation are raised to an odd power, then resulting equations are equivalent
- If both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced)
- Raising both sides to an even power, may make a false statement true:
- Raising both sides to an odd power never makes a false statement true:

.

Homework Problems

- Section: 10.6
- Page: 709
- Problems: Odd: 7 – 57
- MyMathLab Homework Assignment 10.6 for practice
- MyMathLab Quiz 10.6 for grade

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