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Chapter 7 Radicals, Radical Functions, and Rational Exponents

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§ 7.1 Radical Expressions and Functions

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Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.1 Radicals In this section, we introduce a new category of expressions and functions that contain roots. For example, the reverse operation of squaring a number is finding the square root of the number. The symbol that we use to denote the principal square root is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as a radical expression.

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Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.1 Radical ExpressionsEXAMPLE Radical Expression Radicand Radical Sign Index of the Radical

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Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.1 Radical Expressions Definition of the Principal Square Root If a is a nonnegative real number, the nonnegative number b such that, denoted by, is the principal square root of a.

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Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.1 Radical ExpressionsEXAMPLE Evaluate: SOLUTION The principal square root of a negative number, -16, is not a real number. Simplify the radicand. The principal square root of 169 is 13. Take the principal square root of 144, 12, and of 25, 5, and then add to get 17.

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Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.1 Radical FunctionsEXAMPLE For the function, find the indicated function value: SOLUTION Substitute 4 for x in Simplify the radicand and take the square root of 9. Substitute 1 for x in Simplify the radicand and take the square root of 3.

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Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.1 Radical Functions Substitute -1/2 for x in Simplify the radicand and take the square root. Substitute -1 for x in Simplify the radicand. The principal square root of a negative number is not a real number. CONTINUED

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Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.1 Radical Functions - Domain We have seen that the domain of a function f is the largest set of real numbers for which the value of f(x) is defined. Because only nonnegative numbers have real square roots, the domain of a square root function is the set of real numbers for which the radicand is nonnegative. In other words, we only use “allowable” x in the domain of the function. Not allowed for x is any value of x that would cause a negative number under a square root.

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Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.1 Radical Functions - DomainEXAMPLE Find the domain of SOLUTION The domain is the set of real numbers, x, for which the radicand, 3x – 15, is nonnegative. We set the radicand greater than or equal to 0 and solve the resulting inequality. The domain of f is

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Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.1 Radical Functions in ApplicationEXAMPLE Police use the function to estimate the speed of a car, f (x), in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. Use the function to solve the following problem. A motorist is involved in an accident. A police officer measures the car’s skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer believe her? Explain.

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Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.1 Radical Functions in ApplicationSOLUTION Use the given function. CONTINUED Substitute 45 for x. Simplify the radicand. Take the square root. The model indicates that the motorist was traveling at 30 miles per hour at the time of the sudden braking. Since the posted speed limit was 35 miles per hour, the officer should believe that she was not speeding.

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Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.1 Radical Expressions Simplifying T For any real number a, In words, the principal square root of is the absolute value of a. The principal root is the positive root.

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(a) To simplify, first write as an expression that is squared:. Then simplify. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.1 Radical ExpressionsEXAMPLE Simplify each expression: SOLUTION The principal square root of an expression squared is the absolute value of that expression. In both exercises, it will first be necessary to express the radicand as an expression that is squared.

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(b) To simplify, first write as an expression that is squared:. Then simplify. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.1 Radical ExpressionsCONTINUED

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Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.1 Radical Expressions Definition of the Cube Root of a Number The cube root of a real number a is written.

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Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.1 Radical FunctionsEXAMPLE For the function, find the indicated function value: SOLUTION Substitute 13 for x in Simplify the radicand and take the cube root of 27. Substitute 0 for x in Simplify the radicand and take the cube root of 1.

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Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.1 Radical Functions Substitute -63 for x in Simplify the radicand and take the cube root of -125 and then simplify. CONTINUED

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Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.1 Radical Expressions Simplifying T For any real number a, In words, the cube root of any expression is that expression cubed.

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Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.1 Radical ExpressionsEXAMPLE Simplify: SOLUTION Begin by expressing the radicand as an expression that is cubed:. Then simplify. We can check our answer by cubing -5x: By obtaining the original radicand, we know that our simplification is correct.

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Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.1 Radical ExpressionsEXAMPLE Find the indicated root, or state that the expression is not a real number: SOLUTION is not a real number because the index, 8, is even and the radicand, -1, is negative. No real number can be raised to the eighth power to give a negative result such as -1. Real numbers to even powers can only result in nonnegative numbers. because. An odd root of a negative real number is always negative.

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Blitzer, Intermediate Algebra, 5e – Slide #22 Section 7.1 Radical Expressions Simplifying T For any real number a, 1) If n is even, 2) If n is odd,

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Blitzer, Intermediate Algebra, 5e – Slide #23 Section 7.1 Radical ExpressionsEXAMPLE Simplify: SOLUTION Each expression involves the nth root of a radicand raised to the nth power. Thus, each radical expression can be simplified. Absolute value bars are necessary in part (a) because the index, n, is even. if n is even. if n is odd.

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