1. H.Melikian/1100/041 Radicals and Rational Exponents Lecture #2 Dr.Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. H.Melikian/1100/042 Definition of the Principal Square Root v If a is a nonnegative real number, the nonnegative number b such that b 2 = a, denoted by b =  a, is the principal square root of a. In general, if b 2 = a, then b is a square root of a.

3. H.Melikian/1100/043 Square Roots of Perfect Squares For any real number a In words, the principal square root of a 2 is the absolute value of a.

4. H.Melikian/1100/044 The Product Rule for Square Roots v If a and b represent nonnegative real number, then v The square root of a product is the product of the square roots.

5. H.Melikian/1100/045 Text Example v Simplify a.  500b.  6x  3x Solution:

6. H.Melikian/1100/046 The Quotient Rule for Square Roots v If a and b represent nonnegative real numbers and b does not equal 0, then v The square root of the quotient is the quotient of the square roots.

7. H.Melikian/1100/047 Text Example v Simplify: Solution:

8. H.Melikian/1100/048 Example v Perform the indicated operation: 4  3 +  3 - 2  3. Solution:

9. H.Melikian/1100/049 Example v Perform the indicated operation:  24 + 2  6. Solution:

10. H.Melikian/1100/0410 v 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.

11. H.Melikian/1100/0411 What is a conjugate? v Pairs of expressions that involve the sum & the difference of two terms v The conjugate of a+b is a-b v Why are we interested in conjugates? v When working with terms that involve square roots, the radicals are eliminated when multiplying conjugates

12. H.Melikian/1100/0412 Definition of the Principal nth Root of a Real Number v 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.

13. H.Melikian/1100/0413 Finding the nth Roots of Perfect nth Powers

14. H.Melikian/1100/0414 The Product and Quotient Rules for nth Roots v For all real numbers, where the indicated roots represent real numbers,

15. H.Melikian/1100/0415 Definition of Rational Exponents

16. H.Melikian/1100/0416 Example v Simplify 4 1/2 Solution:

17. H.Melikian/1100/0417 Definition of Rational Exponents v The exponent m/n consists of two parts: the denominator n is the root and the numerator m is the exponent. Furthermore,

18. H.Melikian/1100/0418 v If z is positive integer, which of the following is equal to 2 b. 12z c. d. 8z e. 4z a.

19. H.Melikian/1100/0419 POLYNOMIALS: The Degree of ax n. v If a does not equal 0, the degree of ax n is n. v The degree of a nonzero constant is 0. v The constant 0 has no defined degree. A polynomial in x is an algebraic expression of the form a n x n + a n -1 x n -1 + a n -2 x n -2 + … + a 1 n + a 0 where a n, a n -1, a n -2, …, a 1 and a 0 are real numbers. a n != 0, and n is a non-negative integer. The polynomial is of degree n, an is the leading coefficient, and a 0 is the constant term.

20. H.Melikian/1100/0420 Perform the indicated operations and simplify: (-9x 3 + 7x 2 – 5x + 3) + (13x 3 + 2x 2 – 8x – 6) Solution (-9x 3 + 7x 2 – 5x + 3) + (13x 3 + 2x 2 – 8x – 6) = (-9x 3 + 13x 3 ) + (7x 2 + 2x 2 ) + (-5x – 8x) + (3 – 6)Group like terms. = 4x 3 + 9x 2 +(– 13)x + (-3) Combine like terms. = 4x 3 + 9x 2 - 13x – 3 Text Example

21. H.Melikian/1100/0421 The product of two monomials is obtained by using properties of exponents. For example, (-8x 6 )(5x 3 ) = -8·5x 6+3 = -40x 9 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, 3x 4 (2x 3 – 7x + 3) = 3x 4 · 2x 3 – 3x 4 · 7x + 3x 4 · 3 = 6x 7 – 21x 5 + 9x 4. monomial trinomial Multiplying Polynomials

22. H.Melikian/1100/0422 Multiplying Polynomials when Neither is a Monomial v 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 of Outside terms Product of Inside terms Product of Last terms first last inner outer

23. H.Melikian/1100/0423 Multiply: (3x + 4)(5x – 3). Text Example

24. H.Melikian/1100/0424 Multiply: (3x + 4)(5x – 3). Solution (3x + 4)(5x – 3)= 3x·5x + 3x(-3) + 4(5x) + 4(-3) = 15x 2 – 9x + 20x – 12 = 15x 2 + 11x – 12 Combine like terms. first last inner outer FOIL Text Example

25. H.Melikian/1100/0425 The Product of the Sum and Difference of Two Terms v 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.

26. H.Melikian/1100/0426 The Square of a Binomial Sum v The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.

27. H.Melikian/1100/0427 The Square of a Binomial Difference v The square of a binomial difference is first term squared minus 2 times the product of the terms plus last term squared.

28. H.Melikian/1100/0428 Let A and B represent real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Two Terms (A + B)(A – B) = A 2 – B 2 (2x + 3)(2x – 3) = (2x) 2 – 3 2 = 4x 2 – 9 Squaring a Binomial (A + B) 2 = A 2 + 2AB + B 2 (y + 5) 2 = y 2 + 2·y·5 + 5 2 = y 2 + 10y + 25 (A – B) 2 = A 2 – 2AB + B 2 (3x – 4) 2 = (3x) 2 – 2·3x·4 + 4 2 = 9x 2 – 24x + 16 Cubing a Binomial (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3 (x + 4) 3 = x 3 + 3·x 2 ·4 + 3·x·4 2 + 4 3 = x 3 + 12x 2 + 48x + 64 (A – B) 3 = A 3 – 3A 2 B + 3AB 2 - B 3 (x – 2) 3 = x 3 – 3·x 2 ·2 + 3·x·2 2 - 2 3 = x 3 – 6x 2 – 12x + 8

29. H.Melikian/1100/0429 Example v x 2 – y 2 = (x - y)(x + y) v x 2 + 2xy + y 2 = (x + y) 2 v x 2 - 2xy + y 2 = (x - y) 2 v A. if x 2 – y 2 = 24 and x + y = 6, then v x – y = v B. if x – y = 5 and x 2 + y 2 = 13, then v -2xy =

30. H.Melikian/1100/0430 Multiply: a. (x + 4y)(3x – 5y)b. (5x + 3y) 2 Solution We 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) = 3x 2 – 5xy + 12xy – 20y 2 = 3x 2 + 7xy – 20y 2 Combine like terms. (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A 2 + 2AB + B 2 = 25x 2 + 30xy + 9y 2 FOIL Text Example

31. H.Melikian/1100/0431 Example v Multiply: (3x + 4) 2. ( 3x + 4 ) 2 = (3x) 2 + (2)(3x) (4) + 4 2 = 9x 2 + 24x + 16 Solution: