Créer une présentation
Télécharger la présentation

Télécharger la présentation
## Powers, Roots, and Radicals

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Powers, Roots, and Radicals**Algebra 2 Chapter 7 Mr. Hardy**DO NOW**• Evaluate the expression • 1) √9 • 2) -√121 • 3) (√25)2 • Solve each equation • 4) x2 = 49 • 5) (x – 1)2 = 64**Chapter 7 Overview**• Powers, roots, and radicals • How to use rational exponents and nth roots of numbers • How to perform operations with and find inverses of functions • How to graph radical functions and solve radical equations**Chapter 7.1 Lesson Opener**• You will havetwominutes to complete the lesson opener in your notebook. • Evaluate • (2)(2)(2)(2) • (-2)4 • (-2)(-2)(-2) • (2)(2)(2)**nth roots**• For an integer n greater than 1, if bn = a, then b is an nth root of a. • An nth root of a is written: • where nis the index of the radical**DO NOW**• Page 400 • #1 – 13**Notice**• An nth root can also be written as a power of a. • Suppose:**THEREFORE**In General Where n is greater than one (positive)**Finding nth Roots**• Find the indicated real nth root(s) of a • n = 2, a = 16 • n is even, and a > 0; therefore, a has two real nth roots • √16 • √16 = ±4**Example 2**• n = 3, a = -1 • n is odd; therefore, a has one real nth root • Hint: What to the power of 3 is -1?**Example 3**• n = 4, a = -16 • n is even, but a is negative. How many real nth roots?**Try These**• n = 3, a = 125 • n = 3, a = 0 • n = 2, a = 49 • 5 • 0 • 7**Rational Exponents**• Keep in mind: Rational exponents do not have to be in the form of 1/n. Other rational numbers can be used as exponents as well!**Try These**• 9 • 7 • 1/27**Solving Equations**• Example 1 • - 5x2 = -30 • x2 = 6 • x = ±√6 • Use a calculator to round the result • x ≈ ±2.45**Example 2**• (x + 4)3 = 27 • x + 4 = 3 (Take the cube root of both sides) • x = -1**Try These**• x4 = 87 • 2x3 = 92 • (x – 1)5 = 12 • ±3.05 • 3.58 • 2.64**Your Turn**Classwork Homework • Complete Practice B worksheet • Chapter 7.1, page 404 #14-60 even, 65-66 all**Challenge**• Rationalize each denominator, and express the fraction in simplest form**Do Now 3-15-13**• Homework Quiz • Chapter 6.8 • #24, 26, 28 • Chapter 7.1 • #16, 28, 40, 56**Recall**• Rewrite the Expression • 1) • 2) • 3) • Solve the equation. Round if necessary • 2x4 = 35 • 5x3 + 10 = 961**Section 7.2 Opener**• Evaluate the expression • 1) 42 43 • 2) (22)3 • 3) 34/32 • 4) 2-3 • 5) (2 3)4**Using Properties of Rational Exponents**• Example 1 • Same bases- ADD the EXPONENTS**Example 2**• MULTIPLY the EXPONENTS • (21)(x2) = 2x2**Try These**• x1/2 • 256 • y1/3 • 4**Simplest Form**• For a radical to be in simplest form, you must apply the properties of radicals, remove any perfect nth powers (other than 1) and rationalize any denominators. • Two radical expressions are like radicals if they have the same index and the same radicand.**Examples**• √2 * √8 • = √2*8 • =√16 • =4**Writing Radicals and Variable Expression in Simplest Form**• Example 1 • Factor out the perfect fourth root • Use the Product Property**Example 2**• Make the denominator a perfect cube (in order to rationalize**Do Now Looseleaf Silently**Agree or Disagree with the statement that follows and support it with specific and accurate facts that include explanations/proofs, sentences, number statements, and worked out mathematical solutions. The expression can be rewritten as**Recall**• Simplify**Recall: nth roots w/ variables **RECALL from Chapter 5 FACTOR out the PERFECT SQUARE**Application**Use what you know to find a radical expression for the perimeter of the triangle. Simplify the expression. 4ft 1ft 2ft 2ft**The function f(x) = 70x3/4…**• …models the number of calories per day, f(x), a person needs to maintain life in terms of that person's weight, x, in kilograms. (1 kilogram is approximately 2.2 pounds.) Use this model and a calculator to solve.Round answers to the nearest calorie. • How many calories per day does a person who weighs 80 kilograms (approximately 176 pounds) need to maintain life?70 kilograms (approximately 154 pounds)?**Challenge**• Solve the equation**Chapter 7.3Operations on Functions**• Addition: h(x) = f(x) + g(x) • Subtraction: h(x) = f(x) – g(x) • Multiplication: h(x) = f(x)*g(x) or f(x)g(x) • Division: h(x) = f(x)/g(x) or f(x) ÷ g(x) • Composition h(x) = f(g(x)) OR g(f(x))**Power Functions**• y = axb • Where a is a real number and b is a rational number • Note, when b is a positive integer, a power function is simply a type of polynomial function**Note**The rules for thedomainof functions would apply to these combinations of functions as well. The domain of the sum, difference or product would be the numbers x in the domains of both f and g. For the quotient, you would also need to exclude any numbers x that would make the resulting denominator 0.**Addition Example**• f(x) = 3x + 8; g(x) = 2x – 12 • (f + g)(x) = f(x) + g(x) = (3x + 8) + (2x – 12) • What like terms do we have? Combine them and then we have our new function. • (f + g)(x) = 5x - 4**Subtraction Example**• f(x) = 5x2 – 4x; g(x) = 5x + 1 • (f – g)(x) = f(x) – g(x) = (5x2 – 4x) – (5x + 1) • What like terms do we have? Combine them and then we have our new function. • Be careful that you know you are subtracting the whole function g(x) • (f – g)(x) = 5x2 – 9x – 1