Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.2 Radical Functions Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

7.1 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §7.1 → Cube & nth Roots Any QUESTIONS About HomeWork §7.1 → HW-24

Cube Root The CUBE root, c, of a Number a is written as: The number c is the cube root of a, if the third power of c is a; that is; if c3 = a, then

Example  Cube Root of No.s Find Cube Roots a) b) c) SOLUTION a) As 0.2·0.2·0.2 = 0.008 b) As (−13)(−13)(−13) = −2197 As 33 = 27 and 43 = 64, so (3/4)3 = 27/64 c)

Rational Exponents Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow from the Exponent PRODUCT RULE that a1/2a1/2 = a1/2 + 1/2, or a1 Recall  [SomeThing]·[SomeThing] = [SomeThing]2 This suggests that a1/2 is a square root of a.

Definition of a1/n When a is NONnegative, n can be any natural number greater than 1. When a is negative, n must be odd. Note that the denominator of the exponent becomes the index and the BASE becomes the RADICAND.

nth Roots nth root: The number c is an nth root of a number a if cn = a. The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is omitted.

Evaluating a1/n Evaluate Each Expression 27 (a) 271/3 = = 3 64 (b) = 3 64 (b) 641/2 = = 8 625 4 (c) –6251/4 = – = –5 –625 4 (d) (–625)1/4 = is not a real number because the radicand, –625, is negative and the index is even.

Caveat on Roots CAUTION: Notice the difference between parts (c) and (d) in the last Example. The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is NOT a real no. 625 4 (c) –6251/4 = – = –5 –625 4 is not a real number because the radicand, –625, is negative and the index is even. (d) (–625)1/4 =

Radical Functions Given PolyNomial, P, a RADICAL FUNCTION Takes this form: Example  Given f(x) = Then find f(3). SOLUTION To find f(3), substitute 3 for x and simplify.

Example  Exponent to Radical Write an equivalent expression using RADICAL notation a) b) c) SOLUTION a) c) b)

Example  Radical to Exponent Write an equivalent expression using EXPONENT notation a) b) SOLUTION a) b)

Exponent ↔ Index Base ↔ Radicand From the Previous Examples Notice: The denominator of the exponent becomes the index. The base becomes the radicand. The index becomes the denominator of the exponent. The radicand becomes the base.

Definition of am/n For any natural numbers m and n (n not 1) and any real number a for which the radical exists,

Example  am/n Radicals Rewrite as radicals, then simplify a. 272/3 b. 2433/4 c. 95/2 SOLUTION

Example  am/n Exponents Rewrite with rational exponents SOLUTION

Definition of a−m/n For any rational number m/n and any positive real number a the NEGATIVE rational exponent: That is, am/n and a−m/n are reciprocals

Caveat on Negative Exponents A negative exponent does not indicate that the expression in which it appears is negative; i.e.;

Example  Negative Exponents Rewrite with positive exponents, & simplify a. 8−2/3 b. 9−3/2x1/5 c. SOLUTION

Example  Speed of Sound Many applications translate to radical equations. For example, at a temperature of t degrees Fahrenheit, sound travels S feet per second According to the Formula

Example  Speed of Sound During orchestra practice, the temperature of a room was 74 °F. How fast was the sound of the orchestra traveling through the room? SOLUTION: Substitute 74 for t in the Formula and find an approximation using a calculator.

WhiteBoard Work Problems From §7.2 Exercise Set The MACH No. M 4, 10, 18, 32, 48, 54, 130 The MACH No. M

Ernst Mach Fluid Dynamicist All Done for Today Ernst Mach Fluid Dynamicist Born 8Feb1838 in Brno, Austria Died 19Feb1916 (aged 78) in Munich, Germany

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

Graph y = |x| Make T-table