10.1. Square Roots and Square- Root Functions 2/20/2018. Exponents and Radicals. Radical Expressions and Functions
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1 10 Exponents and Radicals 10.1 Radical Expressions and Functions 10.2 Rational Numbers as Exponents 10.3 Multiplying Radical Expressions 10.4 Dividing Radical Expressions 10.5 Expressions Containing Several Radical Terms 10.6 Solving Radical Equations 10.7 The Distance and Midpoint Formulas and Other Applications 10.8 The Complex Numbers 10.1 Radical Expressions and Functions Square Roots and Square-Root Functions Expressions of the Form 2 a Cube Roots Odd and Even nth Roots Square Roots and Square- Root Functions When a number is multiplied by itself, we say that the number is squared. Often we need to know what number was squared in order to produce some value a. If such a number can be found, we call that number a square root of a. Slide 7-3 1
2 Square Root The number c is a square root of a if c 2 = a. Slide 7-4 For example, 16 has 4 and 4 as square roots because ( 4) 2 = 16 and 4 2 = does not have a real-number square root because there is no real number c for which c 2 = 9. Slide 7-5 Find two square roots of 49. Slide 7-6 2
3 Whenever we refer to the square root of a number, we mean the nonnegative square root of that number. This is often referred to as the principal square root of the number. Slide 7-7 Principal Square Root The principal square root of a nonnegative number is its nonnegative square root. The symbol is called a radical sign and is used to indicate the principal square root of the number over which it appears. Slide 7-8 Simplify each of the following. a) b) 81 Slide 7-9 3
4 a is also read as the square root of a, root a, or radical a. Any expression in which a radical sign appears is called a radical expression. The following are examples of radical expressions: 2 3x 2x 12, 3m 2, and 3. 7 The expression under the radical sign is called the radicand. Slide 7-10 Expressions of the Form 2 a 2 It is tempting to write a a, but the next example shows that, as a rule, this is untrue. Slide 7-11 Simplifying 2 a For any real number a, 2 a a. (The principal square root of a 2 is the absolute value of a.) Slide
5 Simplify each expression. Assume that the variable can represent any real number. 2 a) ( y 3) 12 b) m 10 c) x Solution Slide 7-13 Cube Roots We often need to know what number was cubed in order to produce a certain value. When such a number is found, we say that we have found the cube root. For example, 3 is the cube root of 27 because 3 3 =27. Slide 7-14 Cube Root The number c is a cube root of a if c 3 = a. In symbols, we write 3 a to denote the cube root of a. Slide
6 Simplify 3 27 x 3. Slide 7-16 Odd and Even nth Roots The fourth root of a number a is the number c for which c 4 = a. We write n a for the nth root. The number n is called the index (plural, indices). When the index is 2, we do not write it. Slide 7-17 a) b) c) m Simplify each expression. Slide
7 Simplify each expression, if possible. Assume that variables can represent any real number a) 81 b) 81 c) 16m Solution Slide 7-19 Simplifying nth Roots n a n a n a n Even Odd Positive Negative Positive Negative Positive Not a real number Positive Negative a a Slide Rational Numbers as Exponents Rational Exponents Negative Rational Exponents Laws of Exponents Simplifying Radical Expressions 7
8 Rational Exponents Consider a 1/2 a 1/2. If we still want to add exponents when multiplying, it must follow that a 1/2 a 1/2 = a 1/2 + 1/2, or a 1. This suggests that a 1/2 is a square root of a. Slide / n a = n a 1/ n a means n a. When a is nonnegative, n can be any natural number greater than 1. When a is negative, n can be any odd natural number greater than 1. Note that the denominator of the exponent becomes the index and the base becomes the radicand. Slide 7-23 Write an equivalent expression using radical notation and, if possible, simplify. 1/3 a) m 8 1/2 b) 9x 2 1/5 c) xy z Slide
9 Write an equivalent expression using exponential notation and simplify. a) 3 4x 2 5 5y b) x Slide 7-25 Positive Rational Exponents For any natural numbers m and n (n not 1) and any real number a for which n a exists, m/ n m n m a means n a, or a. Slide 7-26 Write an equivalent expression using radical notation and simplify. 2/3 a) 8 3/2 b) 9 Slide
10 Write an equivalent expression using exponential notation. 3 5 a) x 2 b) 5 3xy Slide 7-28 Negative Rational Exponents For any rational number m/n and any m/ n nonzero real number a for which a exists, m/ n 1 a means. m/ n a Slide 7-29 Caution! A negative exponent does not indicate that the expression in which it appears is 1 negative: a a. Slide
11 Write an equivalent expression with positive exponents and, if possible, simplify. 2/3 a) 8 3/2 1/5 b) 9 x 3/4 3t c) 2r Slide 7-31 Laws of Exponents The same laws hold for rational exponents as for integer exponents. Slide 7-32 Laws of Exponents For any real numbers a and b and any rational exponents m and n for which a m, a n, and b m are defined: 1. m n m n a a a m a m n a n a m n mn a a ab m a m b m ( a 0) Slide
12 Use the laws of exponents to simplify 2/5 1/5 a) 7 7 1/2 m b) 1/4 m 1/2 1/3 3/4 c) x y Slide 7-34 Simplifying Radical Expressions Many radical expressions contain radicands or factors of radicands that are powers. When these powers and the index share a common factor, rational exponents can be used to simplify the expression. Slide 7-35 To Simplify Radical Expressions 1. Convert radical expressions to exponential expressions. 2. Use arithmetic and the laws of exponents to simplify. 3. Convert back to radical notation as needed. Slide
13 Use rational exponents to simplify. Do not use exponents that are fractions in the final answer. 4 2 a) (3 x) b) xy z c) 4 y Slide Multiplying Radical Expressions Multiplying Radical Expressions Simplifying by Factoring Multiplying and Simplifying Multiplying Radical Expressions Note that This example suggests the following. Slide
14 The Product Rule for Radicals For any real numbers na and nb, na nb na b. (The product of two nth roots is the nth root of the product of the two radicands.) Rational exponents can be used to derive this rule: n 1/ n 1/ n n a n b a b a b n a b. 1/ Slide 7-40 Multiply. a) 5 6 b) c) 4 x z Slide 7-41 Caution! The product rule for radicals applies only when radicals have the same index: na mb nma b. Slide
15 Simplifying by Factoring The number p is a perfect square if there exists a rational number q for which q 2 = p. We say that p is a perfect nth power if q n = p for some rational number q. The product rule allows us to simplify n ab whenever ab contains a factor that is a perfect nth power. Slide 7-43 Using The Product Rule to Simplify n ab n a n b. ( n a and n b must both be real numbers.) Slide 7-44 To Simplify a Radical Expression with Index n by Factoring 1. Express the radicand as a product in which one factor is the largest perfect nth power possible. 2. Take the nth root of each factor. 3.Simplify the expression containing the perfect nth power. 4. Simplification is complete when no radicand has a factor that is a perfect nth power. Slide
16 It is often safe to assume that a radicand does not represent a negative number raised to an even power. We will make this assumption. Slide 7-46 Simplify by factoring: a) 300 b) 4 8mn 3 4 c) 54s Slide 7-47 Remember: To simplify an nth root, identify factors in the radicand with exponents that are multiples of n. Slide
17 Multiplying and Simplifying We have used the product rule for radicals to find products and also to simplify radical expressions. For some radical expressions, it is possible to do both: First find a product and then simplify. Slide 7-49 Multiply and simplify. a) b) 9x y 9x y Slide Dividing Radical Expressions Dividing and Simplifying Rationalizing Denominators or Numerators With One Term 17
18 Dividing and Simplifying Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots. Slide 7-52 The Quotient Rule for Radicals For any real numbers na and nb, b 0, a n a n. b n b Remember that an nth root is simplified when its radicand has no factors that are perfect nth powers. Recall too that we assume that no radicands represent negative quantities raised to an even power. Slide 7-53 Simplify by taking roots of the numerator and denominator. 49 a) 4 x m b) 3 n Slide
19 Divide and, if possible, simplify. 48 a) x y b) 2 3 3x Slide 7-55 Rationalizing Denominators or Numerators With One Term When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator no longer contains a radical. The procedure for finding such an expression is called rationalizing the denominator. We carry this out by multiplying by 1 in either of two ways. Slide 7-56 One way is to multiply by 1 under the radical to make the denominator of the radicand a perfect power. Slide
20 Rationalize each denominator. a) b) Slide 7-58 Another way to rationalize a denominator is to multiply by 1 outside the radical. Slide a) 3x 3y b) 3 2 4xy Rationalize each denominator. Slide
21 Sometimes in calculus it is necessary to rationalize a numerator. To do so, we multiply by 1 to make the radicand in the numerator a perfect power. Slide Expressions Containing Several Radical Terms Adding and Subtracting Radical Expressions Products and Quotients of Two or More Radical Terms Rationalizing Denominators or Numerators With Two Terms Terms with Differing Indices Adding and Subtracting Radical Expressions When two radical expressions have the same indices and radicands, they are said to be like radicals. Like radicals can be combined (added or subtracted) in much the same way that we combined like terms earlier in this text. Slide
22 Simplify by combining like radical terms. a) b) 7 9s 9s 2 9s Slide 7-64 Simplify by combining like radical terms. a) b) 10 3m 24m Slide 7-65 Products and Quotients of Two or More Radical Terms Radical expressions often contain factors that have more than one term. Multiplying such expressions is similar to finding products of polynomials. Some products will yield like radical terms, which we can now combine. Slide
23 Multiply. Simplify if possible. a) 2( y 7) x x m n m n b) 2 3 c) Slide 7-67 In part (c) of the last example, notice that the inner and outer products in FOIL are opposites, the result, m n, is not itself a radical expression. Pairs of radical terms like, m n and m n, are called conjugates. Slide 7-68 Rationalizing Denominators or Numerators With Two Terms The use of conjugates allows us to rationalize denominators or numerators with two terms. Slide
24 Rationalize the denominator: 5. 7 y Slide 7-70 To rationalize a numerator with more than one term, we use the conjugate of the numerator. Slide 7-71 Terms with Differing Indices To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation. Slide
25 To Simplify Products or Quotients with Differing Indices 1. Convert all radical expressions to exponential notation. 2. When the bases are identical, subtract exponents to divide and add exponents to multiply. This may require finding a common denominator. 3. Convert back to radical notation and, if possible, simplify. Slide 7-73 Multiply and, if possible, simplify: Solution 5 x x 3. Converting to exponential notation Adding exponents Converting to radical notation Simplifying Slide Solving Radical Equations The Principle of Powers Equations with Two Radical Terms 25
26 The Principle of Powers A radical equation is an equation in which the variable appears in a radicand. s are 4 5x and m 2 m 9. Slide 7-76 The Principle of Powers If a = b, then a n = b n for any exponent n. Slide 7-77 Solution Solve: m 3 9. Slide
27 Caution! Raising both sides of an equation to an even power may not produce an equivalent equation. In this case, a check is essential. Slide 7-79 To Solve an Equation with a Radical Term 1. Isolate the radical term on one side of the equation. 2. Use the principle of powers and solve the resulting equation. 3. Check any possible solution in the original equation. Slide 7-80 Solve: x x 5 1. Solution Slide
28 Solution Solve: 3x 4 1/ Slide 7-82 Equations with Two Radical Terms A strategy for solving equations with two or more radical terms is as follows. Slide 7-83 To Solve an Equation with Two or More Radical Terms 1. Isolate one of the radical terms. 2. Use the principle of powers. 3. If a radical remains, perform steps (1) and (2) again. 4. Solve the resulting equation. 5. Check the possible solutions in the original equation. Slide
29 Solve: x 3 x 2 1. Solution Slide The Distance and Midpoint Formulas and Other Applications Using the Pythagorean Theorem Two Special Triangles The Distance and Midpoint Formulas Using the Pythagorean Theorem There are many kinds of problems that involve powers and roots. Many also involve right triangles and the Pythagorean theorem. Slide
30 The Pythagorean Theorem In any right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a 2 + b 2 = c 2. Hypotenuse c a Leg b Leg Slide 7-88 The Principle of Square Roots For any nonnegative real number n, If x 2 = n, then x n or x n. Slide 7-89 How long is a guy wire if it reaches from the top of a 14-ft pole to a point on the ground 8 ft from the pole? Slide
31 Two Special Triangles When both legs of a right triangle are the same size, we call the triangle an isosceles right triangle. If one leg of an isosceles right triangle has length a then c a 2. c 45 a 45 a Slide 7-91 Lengths Within Isosceles Right Triangles The length of the hypotenuse in an isosceles right triangle is the length of a leg times 2. a 45o a 45 o a 2 Slide 7-92 The hypotenuse of an isosceles right triangle is 8 ft long. Find the length of a leg. Give an exact answer and an approximation to three decimal places. Slide
32 A second special triangle is known as a 30 o -60 o -90 o right triangle, so named because of the measures of its angles. Slide 7-94 Lengths Within 30 o 60 o 90 o Right Triangles The length of the longer leg in a 30 o -60 o -90 o right triangle is the length of the shorter leg times 3. The hypotenuse is twice as long as the shorter leg. a 3 30 o 2a 60 o a Slide 7-95 The shorter leg of a 30 o -60 o -90 o right triangle measures 12 in. Find the lengths of the other sides. Give exact answers and, where appropriate, an approximation to three decimal places. Slide
33 The Distance Formula The distance between the points (x 1, y 1 ) and (x 2, y 1 ) on a horizontal line is x 2 x 1. Similarly, the distance between the points (x 2, y 1 ) and (x 2, y 2 ) on a vertical line is y 2 y 1. Slide 7-97 Now consider any two points (x 1, y 1 ) and (x 2, y 2 ). If x x and y y, these points, along with (x 2, y 1 ), describe a right triangle. The lengths of the legs are x 2 x 1 and y 2 y 1. Slide 7-98 We find d, the length of the hypotenuse, by using the Pythagorean theorem: d 2 = x 2 x y 2 y 1 2. Since the square of a number is the same as the square of its opposite, we can replace the absolute-value signs with parentheses: d 2 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Slide
34 The Distance Formula The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) is given by 2 2 d ( x2 x1) ( y2 y1). Slide Find the distance between (3, 1) and (5, 6). Find an exact answer and an approximation to three decimal places. Slide The Midpoint Formula The distance formula is needed to verify a formula for the coordinates of the midpoint of a segment connecting two points. Slide
35 The Midpoint Formula If the endpoints of a segment are (x 1, y 1 ) and (x 2, y 2 ), then the coordinates of the midpoint are y (x 2, y 2 ) x1 x2 y, 1 y (x 1, y 1 ) x (To locate the midpoint, average the x-coordinates and average the y-coordinates.) Slide Find the midpoint of the segment with endpoints (3, 1) and (5, 6). Slide The Complex Numbers Imaginary and Complex Numbers Addition and Subtraction Multiplication Conjugates and Division Powers of i 35
36 Imaginary and Complex Numbers Negative numbers do not have square roots in the real-number system. A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. The complexnumber system makes use of i, a number that is, by definition, a square root of 1. Slide The Number i i is the unique number for which and i 2 = 1. i 1 We can now define the square root of a negative number as follows: p 1 p i p or pi, for any positive number p. Slide Express in terms of i: a) 9 b) 48 Slide
37 Imaginary Numbers An imaginary number is a number that can be written in the form a + bi, where a and b are real numbers and b 0. Slide The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers. Complex Numbers A complex number is any number that can be written in the form a + bi, where a and b are real numbers. (Note that a and b both can be 0.) Slide The following are examples of imaginary numbers: 7 2i 1 2 i 3 11i Slide
38 Slide Addition and Subtraction The complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials. Slide Add or subtract and simplify. a) (3 2 i) (7 8 i) b) (10 2 i) (9 i) Slide
39 Multiplication To multiply square roots of negative real numbers, we first express them in terms of i. For example, i 6 i 5 i Slide Caution! With complex numbers, simply multiplying radicands is incorrect when both radicands are negative: Slide Multiply and simplify. When possible, write answers in the form a + bi. a) 2 10 i i i i b) c) Slide
40 Conjugate of a Complex Number The conjugate of a complex number a + bi is a bi, and the conjugate of a bi is a + bi. Slide Find the conjugate of each number. a) 4 3i b) 6 9i c) i Slide Conjugates and Division Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. Slide
41 Divide and simplify to the form a + bi. 3 2i a) i 4 i b) 2 3 i Slide Powers of i Simplifying powers of i can be done by using the fact that i 2 = 1 and expressing the given power of i in terms of i 2. Consider the following: i 23 = (i 2 ) 11 i 1 = ( 1) 11 i = i Slide Simplify: 40 a) i ; 33 b) i. Solution Slide
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