h.melikian/1100/041 radicals and rational exponents lecture #2 dr.hayk melikyan departmen of...

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Radicals and Rational Exponents

Lecture #2

Dr .Hayk MelikyanDepartmen of Mathematics and CS

melikyan@nccu.edu

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Definition of the Principal Square Root

If a is a nonnegative real number, the nonnegative number b such that b2 = a,

denoted by b = a, is the principal square root of a.

In general, if b2 = a, then b is a square root of a.

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Square Roots of Perfect Squares

a2 a

For any real number a

In words, the principal square root of a2 is the absolute value of a.

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The Product Rule for Square Roots

If a and b represent nonnegative real number, then

The square root of a product is the product of the square roots.

ab a b and a b ab

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Text Example

Simplify a. 500 b. 6x3x

Solution:

b. 6x 3x 6x3x

18x2 9x2 2

9x2 2 9 x2 2

3x 2

a. 500 100 5

100 5

10 5

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The Quotient Rule for Square Roots

If a and b represent nonnegative real numbers and b does not equal 0, then

The square root of the quotient is the quotient of the square roots.

a

ba

band

a

b

a

b.

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Text Example

Simplify:

Solution:

100

9

100

9

10

3

100

9

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3332334

Example

Perform the indicated operation:

43 + 3 - 23.

Solution:

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Example

Perform the indicated operation:

24 + 26.

Solution:

646262

6224

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Rationalizing the denominator: If the denominator contains the square

root of a natural number that is not a perfect

square, multiply the numerator and denominator

by the smallest number that produces the

square root of a perfect square in the denominator.

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What is a conjugate?

Pairs of expressions that involve the sum & the difference of two terms

The conjugate of a+b is a-b Why are we interested in conjugates? When working with terms that involve

square roots, the radicals are eliminated when multiplying conjugates

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Definition of the Principal nth Root of a Real Number

If n, the index, is even, then a is nonnegative (a > 0) and b is also nonnegative (b > 0) . If n is odd, a and b can be any real numbers.

an b means that bn a

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Finding the nth Roots of Perfect nth Powers

If n is odd , ann a

If n is even ann a .

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The Product and Quotient Rules for nth Roots

For all real numbers, where the indicated roots represent real numbers,

an bn abn andan

bn

a

bn , b 0

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Definition of Rational Exponents

a1 / n an .

Furthermore,

a 1/ n 1a1/ n

1an

, a 0

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2442

1

Example

Simplify 4 1/2

Solution:

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Definition of Rational Exponents

The exponent m/n consists of two parts: the denominator n is the root and the numerator m is the exponent. Furthermore,

a m / n 1

am / n .

am / n ( an )m amn .

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If z is positive integer, which of the following is equal to 2

z322

z162

b. 12zc. z8

2

d. 8ze. 4z

a.

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POLYNOMIALS: The Degree of axn.

If a does not equal 0, the degree of axn is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.

A polynomial in x is an algebraic expression of the form

anxn + an-1x

n-1 + an-2xn-2 + … + a1n + a0

where an, an-1, an-2, …, a1 and a0 are real numbers.

an != 0, and n is a non-negative integer.

The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.

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Perform the indicated operations and simplify:(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)

Solution(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)= (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms.= 4x3 + 9x2 +(– 13)x + (-3) Combine like terms.= 4x3 + 9x2 - 13x – 3

Text Example

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The product of two monomials is obtained by using properties of exponents. For example,

(-8x6)(5x3) = -8·5x6+3 = -40x9

Multiply coefficients and add exponents.

Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For example,

3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 + 9x4.monomial trinomial

Multiplying Polynomials

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Multiplying Polynomials when Neither is a Monomial

Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.

Using the FOIL Method to Multiply Binomials

(ax + b)(cx + d) = ax · cx + ax · d + b · cx + b · d

Product of

First terms

Product ofOutside terms

Product ofInside terms

Product of

Last terms

firstlast

inner

outer

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Multiply: (3x + 4)(5x – 3).

Text Example

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Multiply: (3x + 4)(5x – 3).

Solution 

(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3)= 15x2 – 9x + 20x – 12= 15x2 + 11x – 12 Combine like terms.

firstlast

inner

outer

F O I L

Text Example

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The Product of the Sum and Difference of Two Terms

(A B)(A B) A2 B2

The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term.

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The Square of a Binomial Sum

The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.

(A B)2 A2 2AB B2

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The Square of a Binomial Difference

The square of a binomial difference is first term squared minus 2 times the product of the terms plus last term squared.

(A B)2 A2 2AB B2

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Let A and B represent real numbers, variables, or algebraic expressions. 

Special Product ExampleSum and Difference of Two Terms(A + B)(A – B) = A2 – B2 (2x + 3)(2x – 3) = (2x) 2 – 32

= 4x2 – 9

Squaring a Binomial(A + B)2 = A2 + 2AB + B2 (y + 5) 2 = y2 + 2·y·5 + 52

= y2 + 10y + 25(A – B)2 = A2 – 2AB + B2 (3x – 4) 2 = (3x)2 – 2·3x·4 + 42

= 9x2 – 24x + 16

Cubing a Binomial(A + B)3 = A3 + 3A2B + 3AB2 + B3 (x + 4)3 = x3 + 3·x2·4 + 3·x·42 + 43

= x3 + 12x2 + 48x + 64(A – B)3 = A3 – 3A2B + 3AB2 - B3 (x – 2)3 = x3 – 3·x2·2 + 3·x·22 - 23

= x3 – 6x2 – 12x + 8

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Example

x2 – y2 = (x - y)(x + y) x2 + 2xy + y2 = (x + y)2

x2 - 2xy + y2 = (x - y)2

A. if x2 – y2 = 24 and x + y = 6, then x – y =

B. if x – y = 5 and x2 + y2 = 13, then

-2xy =

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Multiply: a. (x + 4y)(3x – 5y) b. (5x + 3y) 2

SolutionWe will perform the multiplication in part (a) using the FOIL method. We will multiply in part (b) using the formula for the square of a binomial, (A + B) 2. a. (x + 4y)(3x – 5y) Multiply these binomials using the FOIL method.

= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y) = 3x2 – 5xy + 12xy – 20y2

= 3x2 + 7xy – 20y2 Combine like terms.

• (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A2 + 2AB + B2

= 25x2 + 30xy + 9y2

F O I L

Text Example

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Example

Multiply: (3x + 4)2.

( 3x + 4 )2 =(3x)2 + (2)(3x) (4) + 42 =9x2 + 24x + 16

Solution: