Mini Lecture 9.1 Finding Roots
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1 Mini Lecture 9. Finding Roots. Find square roots.. Evaluate models containing square roots.. Use a calculator to find decimal approimations for irrational square roots. 4. Find higher roots. Evaluat. a Given the equation, y =, solve for y given that: 4 a. a. a = 9 a = a = 49 d Use a calculator to approimate each epression and round to three decimal places. If the epression is not a real number, so stat. a. 0 d. 8 Find the indicated root, or state that the epression is not a real number. 4. a. 4 8 d. 4 f. 4 g. 4 h. The symbol is called the 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. The symbol is used to denote the negative square root of a number. The square root of a negative number is not a real number. This also applies to any even 4 8 root (,,...). Not all radicals are square roots. Answers:. a. 0 d.. a.. a d. not a real number 4. a. not a real number d. f. 4 g. h. ML-
2 Mini Lecture 9. Multiplying and Dividing Radicals. Multiply square roots.. Simplify square roots.. Use the quotient rule for square roots. 4. Use the product and quotient rules for other roots.. Multiply using the product rul a. 0 d.. Simplify using the product rul a d. 0. Simplify. (Look for a pattern) Assume all variables represent positive number only. 4 a. d. f. g. 8 h. 9 i. 0 j. k. 4. Simplify. a. 80y d. 4 4 y. Multiply. Then simplify if possibl a. 0 8 d. 4 4 y y. Simplify a. d Simplify. a d. ML-8 Have students memorize perfect square numbers through and perfect cubes through. Get as much out of the radicand as possibl Since radicals are unfamiliar to most students, it is important they see the relationship of squaring numbers and square roots, cubing numbers and cube roots, et Answers:. a. 0 d.. a. 0 4 d. 0. a. d. 4. a. 4 4y d. y 4. a. 9 f. d. y y 0 g. 4 h. 4 i. j. k.. a. 4 d.. a. 4 d. 8
3 Mini Lecture 9. Operations with Radicals. Add and subtract radicals.. Multiply radical epressions with more than one term.. Multiply conjugates. Add or subtract as indicated.. a d Multiply.. a. ( + ) ( + )( 4 + ) ( 9 + )( 4 ) d. ( 4 + )( 4 ) ( )( + ) f. ( + ) Two or more square roots can be combined using the distributive property provided they have the same radicand. In some cases, radicals can be combined after they have been simplified. When multiplying radical epressions, distribut This is similar to multiplying a monomial by a polynomial. When multiplying radical epressions use the FOIL method like multiplying binomials. When multiplying conjugates (epressions that involve the sum and difference of the same two terms), the FOIL method may be used or the special product formula. When using the FOIL method with conjugates the OI (outside & inside) will equal 0. Answers:. a. 4 d. cannot be combined. a d. 4 f. + + ML-9
4 Mini Lecture 9.4 Rationalizing the Denominator. Rationalize denominators containing one term.. Rationalize denominators containing two terms. Multiply and simplify.. a. 4 4 d f. 4 g. 4 4 Rationalize each denominator.. a. f. 0 g. d. h. 8 State the conjugate of each of the following.. a. 4 + d. + Multiply. 4. a. ( + )( ) ( + )( ) ( + )( ) Rationalize each denominator and write in simplest form.. a. d Remind students of the definition of a rational number. This will help them understand the meaning of rationalizing the denominator. It may be helpful to discuss the special product of ( a + b)( a b) with several eamples to let students see again what happens to the middle term when the binomials are foiled. Answers:. a. 4 d.. f. g.. a. f. 0 g. 4. a. a. h. d.. a d d. 4 4 ML-0
5 Mini Lecture 9. Radical Equations. Solve radical equations.. Solve problems involving square root models. Solve each radical equation. If the equation has no solution, so stat. a. 4 + = + = = 0 d. + = = + f. + = g. + = 4 h. + 4 = i. = j. = + A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. To solve a radical equation containing square roots, first arrange terms so that one radical is isolated on one side of the equation. Net, square both sides of the equation to eliminate the square root. Solve and ALWAYS check the answer in the original equation. There may be an etra solution(s) that does not check in the original equation. This solution(s) is/are called etraneous solutions. Answers:. a. = = = d. no solution i. 8 j., f. no solution g. 0 h. no solution ML-
6 Mini Lecture 9. Rational Eponents. Evaluate epressions with rational eponents.. Solve problems using models with rational eponents. Write each of the following in radical form first, then simplify.. a. 4 d. ( ) 8 4. a. 4 4 d. 4 Simplify.. a. d. 8 8 f a d. 4 4 f. n If a graphing calculator is being used in the class, it is helpful to show that is the same as the n using number values. Stress to students that the denominator of a rational eponent is the inde of the corresponding radical epression. When the numerator of a rational eponent is not, the numerator is the power to which the radical is raised. It is usually easier to simplify it this way, but it is possible to raise the radicand to the power instead. When the eponent is negative, write the base as its reciprocal, and raise to the positive power. Answers:. a. 4 = 8 = = d. = 4 8 =. a. ( ) 4 4 = ( ) = ( ) = d. ( ) = 4 ( ) 4 8 d. f. 4. a =. a. d. 4 4 f. ML-
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