Rational Exponents Connection: Relating Radicals and Rational Exponents. Understanding Real Numbers and Their Properties

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1 Name Class 6- Date Rational Exponents Connection: Relating Radicals and Rational Exponents Essential question: What are rational and irrational numbers and how are radicals related to rational exponents? N-RN.. Engage Understanding Real Numbers and Their Properties The Venn diagram shows the relationship between the set of real numbers and its subsets. Real Numbers Irrational Numbers Rational Numbers Integers -5 - π - Whole Numbers p A rational number can be expressed in the form q where p and q are integers and q 0. The decimal form a rational number either terminates or repeats. For instance, _ 4 = 0.75 and - _56 = An irrational number cannot be written as the quotient of two integers, and its decimal form is nonrepeating and nonterminating. For instance, the decimal form of, , neither repeats nor terminates. Real numbers, regardless of whether they are rational or irrational, have the following properties with respect to addition and multiplication. Properties of Real Numbers Commutative Property of Addition a+b=b+a Associative Property of Addition (a + b) + c = a + (b + c) Additive Identity The additive identity is 0, because a + 0 = a. Additive Inverse The additive inverse of a is a, because a + (-a) = 0. Commutative Property of Multiplication a b=b a Associative Property of Multiplication (a b) c = a (b c) Multiplicative Identity The multiplicative identity is, because (a) = a. Multiplicative Inverse The multiplicative inverse of a for a 0 is a, because a ( a ) =. Distributive Property a(b + c) = ab + ac Chapter 6 7 Lesson

2 A set of numbers is closed under an operation if the result of the operation on any two numbers in the set (provided the operation is defined for those two numbers) is another number in that set. To prove that a set is not closed under an operation, you need to find only one counterexample. Closure of Sets Under the Four Basic Operations Set Addition Subtraction Multiplication Division Real numbers Yes Yes Yes Yes Irrational numbers No No No No Rational numbers Yes Yes Yes Yes Integers Yes Yes Yes No REFLECT a. Give a counterexample to show why the set of integers is not closed under division. b. Give a counterexample to show why the set of irrational numbers is not closed under multiplication. c. State the operations under which the set of whole numbers is closed. Then state the operations under which the set of whole numbers is not closed, and give counterexamples to show why. N-RN.. EXAMPLE Proving That a Set Is Closed Given that the set of integers is closed under addition and multiplication, prove that the set of rational numbers is closed under addition. Let a and b be rational numbers. a + b = p q + r _ s = s_ s ( p q ) + q q ( r _ s ) = + Multiply. p, q, r, and s are integers with q and s nonzero. Find a common denominator. = Add the numerators. ps + qr Because ps + qr and qs are integers, qs is a rational number. Chapter 6 8 Lesson

3 REFLECT a. How do you know that ps + qr and qs are integers? ps + qr b. Why does a + b = qs prove that the set of rational numbers is closed under addition? c. Given that the set of rational numbers is closed under addition, how can you prove that the set of rational numbers is closed under subtraction? An indirect proof starts by assuming that what you want to prove is not true. If the assumption leads to a contradiction, then the original statement must be true. N-RN.. EXAMPLE Proving That the Sum of a Rational Number and an Irrational Number Is Irrational Given that the set of rational numbers is closed under addition, prove that the sum of a rational number and an irrational number is an irrational number. Let a be a rational number, let b be an irrational number, and let a + b = c. Assume that c is rational. Rewrite a + b = c as b = -a + c by adding -a to both sides. Because -a and c are both and the set of rational numbers is closed under addition, -a + c must be, which means that b is. This contradicts the condition that b be irrational. So, the assumption that c is rational must be false, which means that c, the sum of a rational number and an irrational number, is. REFLECT a. Indirect proof is also called proof by contradiction. What is the contradiction in the preceding proof? b. Compare an indirect proof to a counterexample. Chapter 6 9 Lesson

4 4 N-RN.. ENGAGE Understanding Radicals and Rational Exponents A radical expression is an expression that is written using the radical sign,. A radical expression has an index and a radicand as identified below. Index (a positive integer) n a Radicand (a nonnegative number when n is even; not restricted when n is odd) Read the expression as the nth root of a. It represents the number whose nth power is a. (When n is even, a positive number a has two nth roots, one positive and one negative, and n a represents the positive nth root.) When the index is not shown, it is understood to be, and the radical is a square root. For example, the positive square root of 5, written 5, represents 5 because 5 = 5. If the index is, then the root is called a cube root. For example, the cube root of -8, written -8, is - because (-) = -8. You can write a radical as a power by extending the properties of integer exponents. For instance, you can write a as a power, a k, as follows: ( a ) = a Definition of square root ( a k ) = a Substitute a k for a. a k = a Power of a power property k = Equate exponents. k = Solve for k. So, a = a _. This result can be generalized to any nth root of a and any nth root of a power of a. Converting Between Radical and Rational Exponent Form If the nth root of a is a real number and m is an integer, then REFLECT n a = a n and n a m = a m n. 4a. Explain why it makes sense that a = a and a = a. Chapter 6 0 Lesson

5 If radical expressions are rewritten in rational exponent form, you can then apply the following properties to simplify them. Properties of Rational Exponents Let a and b be real numbers and m and n be integers. Product of Powers Property a m a n = a m+n Quotient of Powers Property a m a n = a m - n, a 0 Power of a Product Property (a b ) n = a n b n Power of a Quotient Property ( a b) n = a n b n, b 0 Power of a Power Property ( a m ) n = a mn Negative Exponent Property a -n = a n, a 0 5 N-RN.. EXAMPLE Using Exponent Properties to Simplify Radical Expressions Simplify each expression. Assume all variables are positive. A (xy) 6 = (xy) 6 Rewrite using a rational exponent. = (xy) Simplify the exponent. = x y Power of a product property B x x = x x Rewrite using rational exponents. C x 4 x = x x 4 = x Product of powers property = x Simplify the exponent. 6 = x Rewrite the expression in radical form. Rewrite using rational exponents. = x Quotient of powers property = x Simplify the exponent. = x Rewrite the expression in radical form. Chapter 6 Lesson

6 REFLECT 5a. In parts B and C, you started with an expression in radical form, converted to rational exponent form, and then converted back to radical form to record the answer. Explain the purpose of each conversion. 5b. Can a b be simplified? Refer to the properties of exponents to support your answer. 5c. Use the properties of exponents to prove that n a n b = n ab. 6 N-RN.. EXAMPLE Simplifying Expressions Involving Rational Exponents (7 x 9 ) = ( ) ( x ) 9 Power of a product property REFLECT = x Power of a power property = x Simplify exponents. = x Evaluate the numerical power. 6a. Show that you get the same simplified form of (7 x 9 ) if you simplify [(7 x 9 ) ]. That is, square 7 x 9 and then raise to the power. 6b. What is the simplified form of (7 x 9 ) -? How is it related to the simplified form of (7 x 9 )? Chapter 6 Lesson

7 PRACTICE. Given that the set of integers is closed under multiplication, prove that the set of rational numbers is closed under multiplication.. Given that the set of rational numbers is closed under multiplication, how can you prove that the set of rational numbers is closed under division?. Given that the set of rational numbers is closed under multiplication, prove that the product of a nonzero rational number and an irrational number is an irrational number. 4. Given that is a rational number and is an irrational number, classify each number below as either rational or irrational. Explain your reasoning. a. + b. - c. ( + ) ( - ) Hint: Use the distributive property to carry out the multiplication. Chapter 6 Lesson

8 Write each radical expression in rational exponent form. Assume all variables are positive d 6. b 7. 4 m Simplify each expression. Assume all variables are positive. 8. y z 9. x 4 y x y 4. x 4 x. (x)( x ). xy x y 5 4. x 6 x 5. 8 x 6 y x y 8 7. (8 x ) 8. ( x 4 y - 4 ) 9. ( 4 x y ) 6 0. (6 a 9 ). ( a 4 b -8 ) - 4. (6 b - ) -. Explain why the expression x is undefined when x < Use the properties of exponents to show that n a b = n a n. b 5. Show that n a m = ( n a ) m. 6. Error Analysis A student simplified the expression x x by writing x x = x x = x + = x 4_ = x. Describe and correct the student s error. 7. In the expression n a m, suppose m is a multiple of n. That is, m = kn where k is an integer. Show how to obtain the simplified form of n a m. If a is a nonzero rational number, is n a m rational or irrational? Explain. Chapter 6 4 Lesson

9 Name Class Date Additional Practice 6- Simplify each expression. All variables represent nonnegative numbers ( ) ( ) 8. ( 4 ) 8 9. Given a cube with volume, you can use the formula 4 to find the perimeter of one of the cube s square faces. Find the perimeter of a face of a cube that has volume 5 m. Chapter 6 5 Lesson

10 Problem Solving. For a pendulum with a length of L meters, the time in seconds that it takes the pendulum to swing back and forth is approximately L. About how long does it take a pendulum that is 9 meters long to swing back and forth?. The Beaufort Scale is used to measure the intensity of tornados. For a tornado with Beaufort number B, the formula v.9b may be used to estimate the tornado s wind speed in miles per hour. Estimate the wind speed of a tornado with Beaufort number 9.. Given a cube whose faces each have area A, the volume of the cube is given by the formula V A. Find the volume of a cube whose faces each have an area of 64 in. 4. At a factory that makes cylindrical cans, the formula r V is used to find the radius of a can with volume V. What is the radius of a can whose volume is 9 cm? 5. Which is the best estimate for the brain mass of a macaw? A 9 g C 5 g B 45 g D 5 g 7. An animal has a body mass given by the expression x 4. Which expression can be used to estimate the animal s brain mass? A B.8x C B.8x B B.8x 4 D B.8x 6. How much larger is the brain mass of a barn owl compared to the brain mass of a cockatiel? F 89 g H 88.8 g G 40. g J 5 g Cockatiel 8 Guam Rail 56 Macaw 65 Barn Owl 96 Sources: Chapter 6 6 Lesson