course Mth 158 008. `* 8
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Given Solution: * * The cube root of 54 is expressed as 54^(1/3). The number 54 factors into 2 * 3 * 3 * 3, i.e., 2 * 3^3. Thus 54^(1/3) = (2 * 3^3) ^(1/3) = 2^(1/3) * (3^3)^(1/3) = 2^(1/3) * 3^(3 * 1/3) = 2^(1/3) * 3^1 = 3 * 2^(1/3), i.e., 3 * cube root of 2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Knowing the property of cubes is necessary for this problem Self-critique Rating: ********************************************* Question: * R.8.18. Simplify the cube root of (3 x y^2 / (81 x^4 y^2) ). Your Solution: Cancel out like terms giving 1/(1/27x^3) ^(1/3) separate 1/ (1/27)^(1/3) * (x^3)^(1/3) 27 factors to 3^3 and multiply 3*1/3 for law of exponents for x, solving 1/(3x) Confidence Assessment: taking step by step if the main thing for me because looking at the problem seems intimidating but once you simplify and write it out, it is not as bad.
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Given Solution: The cube root of (3 x y^2 / (81 x^4 y^2) ) is (3 x y^2 / (81 x^4 y^2) ) ^ (1/3) = (1 / (27 x^3) ) ^(1/3) = 1 / ( (27)^(1/3) * ^x^3^(1/3) ) = 1 / ( (3^3)^(1/3) * (x^3)^(1/3) ) = 1 / ( 3^(3 * 1/3) * x^(3 * 1/3) ) = 1 / (3 * x) = 1 / (3x). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): it gets confusing to remember if the expression is in the denominator, numerator and you have to be careful. Self-critique Rating: ********************************************* Question: * R.8.30. Simplify 2 sqrt(12) - 3 sqrt(27). Answer: factor within giving 12 = 2*2*3 and 27 = 3*3*3 and separate each square root for each factor and simplify giving 2*2-3*3 sqaure root(3)…4-9 sqaure root of 3 Confidence Assessment:
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Given Solution: 2 sqrt(12) - 3 sqrt(27) = 2 sqrt( 2*2*3) - 3 sqrt(3*3*3) = 2 sqrt(2^2 * 3) - 3 sqrt(3^3) = 2 sqrt(2^2) sqrt^3) - 3 sqrt(3^2) sqrt(3) = 2 * 2 - 3 * 3 sqrt(3) = 4 - 9 sqrt(3). Extra Question: What is the simplified form of (2 sqrt(6) + 3) ( 3 sqrt(6)) and how did you get this result? ********************************************* Your solution: (2*sqrt(6) * 3sqrt(6) + 3*3sqrt(6)) and simplifying to 36 +9sqrt(6) Confidence Assessment:
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Given Solution: (2*sqrt(6) +3)(3*sqrt(6)) expands by the Distributive Law to give (2*sqrt(6) * 3sqrt(6) + 3*3sqrt(6)), which we rewrite as (2*3)(sqrt6*sqrt6) + 9 sqrt(6) = (6*6) + 9sqrt(6) = 36 +9sqrt(6). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: * R.8. Expand (sqrt(x) + sqrt(5) )^2 Answer: (Sqrt x + sqrt 5) * (sqrt x + sqrt 5) expanded sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) ) then sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5) so x + 2 sqrt(x) sqrt(5) + 5 Confidence Assessment: this look very hard because of the way it is written on the computer but for me the best thing to do is keep a piece of paper and write down and work it out on paper before typing it into the computer….it looks completely different and can be confusing!
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Given Solution: (sqrt(x) + sqrt(5) )^2 = (sqrt(x) + sqrt(5) ) * (sqrt(x) + sqrt(5) ) = sqrt(x) * (sqrt(x) + sqrt(5) ) + sqrt(5) * (sqrt(x) + sqrt(5) ) = sqrt(x) * sqrt(x) + sqrt(x) * sqrt(5) + sqrt(5) * sqrt(x) + sqrt(5) * sqrt(5) = x + 2 sqrt(x) sqrt(5) + 5. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): You have to know and recognize the rules and follow through otherwise you’ll get the wrong answer Self-critique Rating: ********************************************* Question: * R.8.42. What do you get when you rationalize the denominator of 3 / sqrt(2) and what steps did you follow to get this result? ********************************************* Your solution: Multiply by sqrt of 2 to get rid of the square root in the denominator giving (3 sqrt 2) / 2 Confidence Assessment:
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Given Solution: Starting with 3/sqrt(2) we multiply numerator and denominator by sqrt(2) to get (2*sqrt(2))/(sqrt(2)*sqrt(2)) = (3 sqrt(2) ) /2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: * R.8.48. Rationalize denominator of sqrt(2) / (sqrt(7) + 2) Answer: again multiply the square root of the denominator so multiply by sqrt(7) – 2 Giving [ sqrt(2) * (sqrt(7) - 2 ] / 3 To rationalize the denominator sqrt(7) + 2 we multiply both numerator and denominator by sqrt(7) - 2. We obtain ( sqrt(2) / (sqrt(7) + 2) ) * (sqrt(7) - 2) / (sqrt(7) - 2) = sqrt(2) * (sqrt(7) - 2) / ( (sqrt(7) + 2) * ( sqrt(7) - 2) ) = sqrt(2) * (sqrt(7) - 2) / (sqrt(7) * sqrt(7) - 4) = sqrt(2) * (sqrt(7) - 2 ) / (7 - 4) = sqrt(2) * (sqrt(7) - 2 ) / 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Extra Question: What steps did you follow to simplify (x^3)^(1/6) and what is your result, assuming that x is positive and expressing your result with only positive exponents? ********************************************* Your solution: 3*1/6 giving an exponent of ˝ so x^(1/2) Confidence Assessment:
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Given Solution: * * Express radicals as exponents and use the laws of exponents. (x^3)^(1/6) = x^(3 * 1/6) = x^(1/2). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Right I multiplied the exponents and used the law of exponents to simplify the expression Self-critique Rating: ********************************************* Question: * R.8.60. Simplify 25^(3/2). Answer: 25 is 5^2 so multiply 2* 3/2 giving 5^3 Confidence Assessment:
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Given Solution: 25^(3/2) = (5^2)^(3/2) = 5^(2 * 3/2) = 5^(2 * 3/2) = 5^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: ********************************************* Question: * R.8.72. Simplify and express with only positive exponents: (xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4) ********************************************* Your Solution: Separate and multiply exponents giving: x^(1/4) * y^(1/4) * x^(2 * 1/2) * y^(2 * 1/2) / ( (x^(2 * 3/4) * y^(3/4) ) then keep simplifying x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4) ) then x^(5/4) y^(5/4) / (x^(3/2) y^(3/4) ) x^(5/4 - 3/2) y^(5/4 - 3/4) = y^(1/2) / x^(1/4) Confidence Assessment:
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Given Solution: (xy)^(1/4) (x^2 y^2) ^(1/2) / (x^2 y)^(3/4) = x^(1/4) * y^(1/4) * (x^2)^(1/2) * y^2 ^ (1/2) / ( (x^2)^(3/4) * y^(3/4) ) = x^(1/4) * y^(1/4) * x^(2 * 1/2) * y^(2 * 1/2) / ( (x^(2 * 3/4) * y^(3/4) ) = x^(1/4) y^(1/4) * x^1 * y^1 / (x^(3/2) y^(3/4) ) = x^(1 + 1/4) y^(1 + 1/4) / (x^(3/2) y^(3/4) ) = x^(5/4) y^(5/4) / (x^(3/2) y^(3/4) ) = x^(5/4 - 3/2) y^(5/4 - 3/4) = x^(5/4 - 6/4) y^(2/4) = x^(-1/4) y^(1/2) = y^(1/2) / x^(1/4). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Must be careful multiplying the exponents and simplifying Self-critique Rating: ********************************************* Question: * R.8.108. v = sqrt(64 h + v0^2); find v for init vel 0 height 4 ft; for init vel 0 and ht 16 ft; for init vel 4 ft / s and height 2 ft. ********************************************* Your Solution: Substitute the variables giving v=sqrt(64 * 4+ o^2 which is sqrt of 256 equals 16 Substitute height of 16 and get sqrt 1024 giving 32 so With initial velocity of 4 ft/s and height of 2 ft - v = sqrt(64 * 2 + 4^2) = sqrt(144) =12 Confidence Assessment: splitting up the word problem and doing each step separately then putting it together is best for me
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Given Solution: If initial velocity is 0 and height is 4 ft then we substitute v0 = 0 and h = 4 to obtain • v = sqrt(64 * 4 + 0^2) = sqrt(256) =16. If initial velocity is 0 and height is 16 ft then we substitute v0 = 0 and h = 4 to obtain • v = sqrt(64 * 16 + 0^2) = sqrt(1024) = 32. Note that 4 times the height results in only double the velocity. If initial velocity is 4 ft / s and height is 2 ft then we substitute v0 = 4 and h = 2 to obtain • v = sqrt(64 * 2 + 4^2) = sqrt(144) =12. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Self-critique Rating: Extra Question: What is the simplified form of (24)^(1/3) and how did you get this result? ********************************************* Your solution: Factor 24 to 8*3 so 8^ 1/3 * 3^ 1/3 giving 2*3^(1/3) Confidence Assessment:
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Given Solution: * * (24)^(1/3) = (8 * 3)^(1/3) = 8^(1/3) * 3^(1/3) = 2 * 3^(1/3) ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Again know the rules and recognize when to factor is key Self-critique Rating: ********************************************* Question: Extra Question: What is the simplified form of (x^2 y)^(1/3) * (125 x^3)^(1/3) / ( 8 x^3 y^4)^(1/3) and how did you get this result? ********************************************* Your solution: Group and multiply like terms giving : (x^(2/3)y^(1/3)* (5x)/[ 8^(1/3) * xy(y^(1/3)] Simplify: 5 ( x^(2/3) ) / (2 y) Confidence Assessment:
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Given Solution: * * (x^2y)^(1/3) * (125x^3)^(1/3)/ ( 8 x^3y^4)^(1/3) (x^(2/3)y^(1/3)* (5x)/[ 8^(1/3) * xy(y^(1/3)] (x^(2/3)(5x) / ( 2 xy) 5( x^(5/3)) / ( 2 xy) 5x(x^(2/3)) / ( 2 xy) 5 ( x^(2/3) ) / (2 y) ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I had to think twice with this question because I worked the problem out and was forgetting to factor through all and got stuck. Self-critique Rating: ********************************************* Question: Extra question. What is the simplified form of sqrt( 4 ( x+4)^2 ) and how did you get this result? Your Solution: It is written as [ (4*x+4)^2)]^1/2 . . ? Confidence Assessment: some problems are tough for me because I’m not sure how the problem is stated because of the way it is written on the computer, but I’m getting used to it now.
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Given Solution: We use two ideas in this solution: • sqrt(a b) = sqrt(a) * sqrt(b) and • sqrt(x^2) = | x | To understand why sqrt(x^2) = | x | and not just x consider the following: • Let x = 5. Then sqrt(x^2) = sqrt( 5^2 ) = sqrt(25) = 5, so sqrt(x^2) = x. It is also clear that in this case, | x | = | 5 | = 5, so | x | = x, and we can say that sqrt(x^2) = | x |. • Now let x = -5. We get sqrt(x^2) = sqrt( (-5)^2 ) = sqrt(25) = 5. In this case sqrt(x^2) = 5 but x is not equal to 5, so sqrt(x^2) is not x. However, in this case | x | = | -5 | = 5, so it is the case the sqrt(x^2) = | x |. Using these ideas we get • sqrt( 4 ( x+4)^2 ) = sqrt(4) * sqrt( (x+4)^2 ) = 2 * | x+4 | ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Okay, after reading the solution I can see how to solve the problem now. Self-critique Rating: * Add comments on any surprises or insights you experienced as a result of this assignment. Insights would be that it is easy to get confused so you have to careful especially with division problems. And knowing different option and rules like the last question.