h.melikian/1100/041 radicals and rational exponents lecture #2 dr.hayk melikyan departmen of...
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H.Melikian/1100/04 1
Radicals and Rational Exponents
Lecture #2
Dr .Hayk MelikyanDepartmen of Mathematics and CS
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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: