Transcript x 2/3
Welcome to the Unit 7 Seminar for College Algebra! Theodore Vassiliadis To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here Unit 5 Seminar Agenda • Radical Expressions and Radical Functions • Simplifying and Combining Radical Expressions • Multiplying and Dividing Radical Expressions Radicals • Radicals are roots. The typical radical symbol √ is considered to be a “square root” symbol. This is true IF you see no number in the crook of the symbol. You can also write sqrt if you can’t get the symbol to work. For example √3 = sqrt(3). Examples of Radicals √8 is “the square root of eight” The index is an understood 2 and the radicand is 8. ______ √100a^2b is “the square root of one hundred a squared b” The index is an understood 2 and the radicand is 100a^2b. ____ 3√27c^6 is “the cube root of twenty-seven c to the sixth power” The index is 3 and the radicand is 27c^6. ___ 5√-32 is “the fifth root of negative thirty-two” The index is 5 and the radicand is –32. Radicals • Roots are the same as the denominator of an exponent – x^1/2 = \/x – x^1/3 = \3/x ( note how we write the root in our editor) – x^3/5 = \5/x^3 • So as we can see here, we can interchange the denominators in the exponents with the roots • This also means that we can use the properties of exponents in order to simplify the expressions we have. Terms with rational exponents are related to terms with radicals. Here’s how. ___ am/n = n√am Examples: ___ x2/3 = 3√x2 ____ 2004/7 = 7√2004 ________ _____ (36a^2b^4)1/2 = 2√(36a2b4)1 = 2√36a2b4 Rational exponents are exponents that are rational numbers Rational exponents are fractions. • Example: x^3/2 is a rational exponent. this can be written as \/(x^3) • Example: x^2/3 is a rational exponent. • Let's practice on the following: –\/[36x^18] in order to solve this problem we remember that the square root of a product is the same as the product of the square roots! So we split the above square root to – \/[36] = 6 – \/x^18 = x^18/2 remember that the square root is the 1/2 exponents so we have x^9 • Thus the final result is 6x^9 Try this: • \/[25y^6] Try this: • • • • \/[25y^6] \/25 = 5 since 5*5 gives us 25 \/y^6 = y^3 since y^3*y^3 = y^(3+3) = y^6 Thus the expression \/[25y^6] can be rewritten as 5y^3 Let’s simplify this • \/[36x^18y^6z^10] • • • • • • • in order to solve this problem we will remember that the square root of a product is the same as the product of the square roots! so we split the above square root to \/[36]* \/x^18 * \/y^6 * \/z^10= \/[36] = 6 \/x^18 = x^18/2 remember that the square root is the 1/2 exponents so we have x^9 \/y^6 for the same reason = y^6/2 = y^3 and \/z^10 = z^10/2 = z^5 so the result will be • 6*x^9y^3z^5 Let’s work with negative exponents • Review: x-1 • x-1 = 1/x • x-2 = 1/x2 • x-1/2 = 1/ \/x • To simplify the above we only use the rules of exponents that are also rules of roots Simplify and express answer with positive exponents (x-1/6x)3/2 Apply the power rule and distribute the outside power x-1/6*3/2 * x3/2 Simplify the exponents x-3/12 * x3/2 Reduce the fractions at the exponents x-1/4 * x3/2 Apply the product rule for multiplying exponents with like bases. x-1/4+3/2 = x-1/4+6/4 =x 5/4 Write the following expression as one fraction containing only positive exponents. 4-1/4 + x2/3 First write the first expression with a positive exponent. = 1 + x2/3 we need to find their common denominator to add those 41/4 The LCD is 41/4 so multiply the second expression by 41/4/41/4 = 1__ + x2/3 (41/4) 41/4 41/4 Add the numerators. = 1 + 41/4x2/3 41/4 this is the final answer Factor out the common factor of 2x from 8x4/3 + 12x3/2 Write each rational exponent with the same LCD to make it easier to factor. LCD=6 = 8x8/6 + 12x9/6 Now write each rational exponent as a sum so that it is easier to see that we are factoring out 2x, and not that x = x6/6 thus we factor 2x6/6. = 8x6/6+ 2/6 + 12x6/6 + 3/6 = 4*2x6/6x2/6 + 6*2x6/6x3/6 Now note that we have a common factor 4*2x6/6x2/6 + 6*2x6/6x3/6 Factor out 2x6/6 = 2x6/6(4x2/6 + 6x3/6) Reduce all powers to lowest terms. = 2x(4x1/3 + 6x1/2)