course mth 158 assignment #002002. `query 2
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19:57:55 query R.2.46 (was R.2.36) Evaluate for x = -2, and y = 3 the expression (2x - 3) / y and explan how you got your result.
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RESPONSE --> 2x-3/y = [2(-2)-3]/3 = (-4-3)/3 = -7/3 To obtain my result, I substituted -2 for X and 3 for Y. In doing so, I followed the properties of the arithmetic of quotients, arriving at the answer -7/3 or -2.333...
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19:58:06 ** Starting with (2x-3)/y we substitute x=-2 and y=3 to get (2*(-2) - 3)/3 = (-4-3)/3= -7/3. **
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RESPONSE --> ok
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20:00:35 query R.2. 55 (was R.2.45) Evaluate for x = 3 and y = -2: | |4x| - |5y| | and explan how you got your result.
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RESPONSE --> | |4x| - |5y| | = | |12| - |-10| | = 12-10 = 2 In computing absolute value, negatives are released -- the absolute value of -10 is 10, so the equation was 12-10 = 2
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20:00:43 ** Starting with | | 4x |- | 5y | | we substitute x=3 and y=-2 to get | | 4*3 | - | 5*-2 | | = | | 12 | - | -10 | | = | 12-10 | = | 2 | = 2. **
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RESPONSE --> ok
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20:02:35 query R.2.64 (was R.2.54) Explain what values, if any, must not be present in the domain of the expression (-9x^2 - x + 1) / (x^3 + x)
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RESPONSE --> The values of 1 and -1 must not be present in the domain of the expression (-9x^2 - x + 1) / (x^3 + x) because these values would cause the denominator of the quotient to be 0 -- thus making the equation undefined.
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20:04:30 ** The denominator of this expression cannot be zero, since division by zero is undefined. Since x^3 + x factors into (x^2 + 1) ( x ) we see that x^3 + x = 0 only if x^2 + 1 = 0 or x = 0. Since x^2 cannot be negative x^2 + 1 cannot be 0, so x = 0 is indeed the only value for which x^3 + x = 0. **
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RESPONSE --> I apologize -- I looked at the wrong question. I answered for #62 where the answers were 1 & -1 (b & d). I did have 0 as the answer for this problem.
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20:07:23 query R.2.76 \ 73 (was R.4.6). What is -4^-2 and how did you use the laws of exponents to get your result?
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RESPONSE --> The problem in the book was stated -4^2, in which the answer would be 16, since two negative numbers multiplied equal a positive number. However, you are asking for -4^-2, in which the answer would be: -4^-2 = 1/-4^2 = 1/16 When there is a negative exponent, you inverse the number.
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20:08:40 ** order of operations implies exponentiation before multiplication; the - in front of the 4 is not part of the 4 but is an implicit multiplication by -1. Thus only 4 is raised to the -2 power. -4^(-2) Since a^-b = 1 / (a^b), we have 4^-2 = 1 / (4)^2 = 1 / 16. The - in front then gives us -4^(-2) = - ( 1/ 16) = -1/16. If the intent was to take -4 to the -2 power the expression would have been written (-4)^(-2).**
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RESPONSE --> Got that one wrong -- did not know that the (-) in front of a number implied multiplication by (-1)
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20:17:58 query Extra Problem. What is (3^-2 * 5^3) / (3^2 * 5) and how did you use the laws of exponents to get your result?
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RESPONSE --> (3^-2 * 5^3) / (3^2 * 5) = (3^-2/3^2) * (5^3/5) = 3^-2-2 * 5^3-1 = 3^-4 * 5^2 = 1/3^4 * 5^2 = 5^2/3^4 = 25/81 Laws of exponents: a^m/a^n = a^m-n An inversion was done to correct the negative exponent
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20:18:40 ** (3^(-2)*5^3)/(3^2*5). Grouping factors with like bases we have 3^(-2)/3^2 * 5^3 / 5. Using the fact that a^b / a^c = a^(b-c) we get 3^(-2 -2) * 5^(3-1), which gives us 3^-4 * 5^2. Using a^(-b) = 1 / a^b we get (1/3^4) * 5^2. Simplifying we have (1/81) * 25 = 25/81. **
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RESPONSE --> ok solved a little differently -- same answer
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20:37:15 query R.2.94. Express [ 5 x^-2 / (6 y^-2) ] ^ -3 with only positive exponents and explain how you used the laws of exponents to get your result.
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RESPONSE --> [ 5 x^-2 / (6 y^-2) ] ^ -3 = [(5^-3)(x^6)]/[(6^-3)(y^6)] = [(6^3)(x^6)]/[(5^3)(y^6)] = 216x^6/125y^6 laws of exponents used: (a^m)^n = a^mn (ab)^n = a^n(b^n)
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20:37:58 [ 5 x^-2 / (6 y^-2) ] ^ -3 = (5 x^-2)^-3 / (6 y^-2)^-3, since (a/b)^c = a^c / b^c. This simplifies to 5^-3 (x^-2)^-3 / [ 6^-3 (y^-2)^-3 ] since (ab)^c = a^c b^c. Then since (a^b)^c = a^(bc) we have 5^-3 x^6 / [ 6^-3 y^6 ] . We rearrange this to get the result 6^3 x^6 / (5^3 y^6), since a^-b = 1 / a^b.
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RESPONSE --> ok
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20:43:02 query Extra Problem. Express (-8 x^3) ^ -2 with only positive exponents and explain how you used the laws of exponents to get your result.
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RESPONSE --> (-8 x^3) ^ -2 = (-8^-2)(x^-6) = (-1/8^2)(1/x^6) = -1/64x^6 exponential law: (ab)^n = a^nb^n
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20:47:42 ** ERRONEOUS STUDENT SOLUTION: (-8x^3)^-2 -1/(-8^2 * x^3+2) 1/64x^5 INSTRUCTOR COMMENT:1/64x^5 means 1 / 64 * x^5 = x^5 / 64. This is not what you meant but it is the only correct interpretation of what you wrote. Also it's not x^3 * x^2, which would be x^5, but (x^3)^2. There are several ways to get the solution. Two ways are shown below. They make more sense if you write them out in standard notation. ONE CORRECT SOLUTION: (-8x^3)^-2 = (-8)^-2*(x^3)^-2 = 1 / (-8)^2 * 1 / (x^3)^2 = 1/64 * 1/x^6 = 1 / (64 x^5). Alternatively (-8 x^3)^-2 = 1 / [ (-8 x^3)^2] = 1 / [ (-8)^2 (x^3)^2 ] = 1 / ( 64 x^6 ). **
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RESPONSE --> Process was correct -- however, in a previous problem, I thought that it was understood that if a (-) was in front of a number that you assumed that number was multiplied by (-1). I left the -1 on the top of the equation and brought 8^2 to the denominator.
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20:51:51 query R.2.90 (was R.4.36). Express (x^-2 y) / (x y^2) with only positive exponents and explain how you used the laws of exponents to get your result.
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RESPONSE --> (x^-2 y) / (x y^2) = (x^-2-1)(y^1-2) = (x^-3)(y^-1) = 1/x^3y exponential law: a^m/a^n = a^m-n
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20:52:05 ** (1/x^2 * y) / (x * y^2) = (1/x^2 * y) * 1 / (x * y^2) = y * 1 / ( x^2 * x * y^2) = y / (x^3 y^2) = 1 / (x^3 y). Alternatively, or as a check, you could use exponents on term as follows: (x^-2y)/(xy^2) = x^-2 * y * x^-1 * y^-2 = x^(-2 - 1) * y^(1 - 2) = x^-3 y^-1 = 1 / (x^3 y).**
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RESPONSE --> ok
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21:06:05 query Extra Problem. . Express 4 x^-2 (y z)^-1 / [ (-5)^2 x^4 y^2 z^-5 ] with only positive exponents and explain how you used the laws of exponents to get your result.
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RESPONSE --> 4 x^-2 (y z)^-1 / [ (-5)^2 x^4 y^2 z^-5 ] = 4x^-2 (y^-1)(z^-1)/25x^4(y^2)(z^-5) = 4(x^-2-4)(y^-1-2)(z^-1-5)/25 = 4x^-6y^-3z^-6/25 = 4/25x^6y^3z^6 exponential law: (ab)^n = a^nb^n
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21:08:24 ** Starting with 4x^-2(yz)^-1/ [ (-5)^2 x^4 y^2 z^-5] Squaring the -5 and using the fact that (yz)^-1 = y^1 * z^-1: 4x^-2 * y^-1 * z^-1/ [25 * x^4 * y^2 * z^-5} Grouping the numbers, and the x, the y and the z expression: (4/25) * (x^-2/x^4) * (y^-1/y^2) * (z^-1/z^-5) Simplifying by the laws of exponents: (4/25) * x^(-2-4) * y^(-1-2) * z^(-1+5) Simplifying further: (4/25) * x^-6 * y^-3 * z^4 Writing with positive exponents: 4z^4/ (25x^6 * y^3 ) **
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RESPONSE --> I got it right to this point: (4/25) * x^(-2-4) * y^(-1-2) * z^(-1+5) messed up and forgot that inversing the -5 exponent for z would make it a postive exponent
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21:09:20 query R.2.122 (was R.4.72). Express 0.00421 in scientific notation.
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RESPONSE --> 0.00421 = 4.21 * 10^-3
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21:09:37 ** 0.00421 in scientific notation is 4.21*10^-3. This is expressed on many calculators as 4.21 E-4. **
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RESPONSE --> ok
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21:09:59 query R.2.128 (was R.4.78). Express 9.7 * 10^3 in decimal notation.
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RESPONSE --> 9.7 * 10^3 = 9700
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21:10:04 ** 9.7*10^3 in decimal notation is 9.7 * 1000 = 9700 **
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RESPONSE --> ok
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21:15:40 query R.2.152 \ 150 (was R.2.78) If an unhealthy temperature is one for which | T - 98.6 | > 1.5, then how do you show that T = 97 and T = 100 are unhealthy?
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RESPONSE --> If | T - 98.6 | > 1.5, then if T=97 | 97 - 98.6 | > 1.5 | 1.6 | > 1.5 1.6 > 1.5 -- T of 97 is unhealthy If | T - 98.6 | > 1.5, then if T=100 | 100 - 98.6 | > 1.5 | 1.4 | > 1.5 1.4 < 1.5 -- T of 100 does not fit the equation of an unhealthy temperature
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21:15:58 ** You can show that T=97 is unhealthy by substituting 97 for T to get | -1.6| > 1.5, equivalent to the true statement 1.6>1.5. But you can't show that T=100 is unhealthy, when you sustitute for T then it becomes | 100 - 98.6 | > 1.5, or | 1.4 | > 1.5, giving us 1.4>1.5, which is an untrue statement. **
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RESPONSE --> ok
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