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22:05:04 query R.1.14 (was R.1.6) Of the numbers in the set {-sqrt(2), pi + sqrt(2), 1 / 2 + 10.3} which are counting numbers, which are rational numbers, which are irrational numbers and which are real numbers?
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RESPONSE --> counting numbers=none rational numbers=1 / 2 + 10.3 irrational= -sqrt(2), pi +sqrt(2) real= -sqrt(2), pi +sqrt(2), 1 / 2 + 10.3
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22:05:30 ** Counting numbers are the numbers 1, 2, 3, .... . None of the given numbers are counting numbers
Rational numbers are numbers which can be expressed as the ratio of two integers. 1/2+10.3 are rational numbers. Irrational numbers are numbers which can be expressed as the ratio of two integers. {-sqrt(2)}, pi+sqrt(2) are irrational numbers.. **......!!!!!!!!...................................
RESPONSE --> ok
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22:08:19 query R.1.32 (was R.1.24) Write in symbols: The product of 2 and x is the product of 4 and 6
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RESPONSE --> 2x=4(6)
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22:09:47 ** The product of 2 and x is 2 * x and the product of 4 and 6 iw 4 * 6. To say that these are identical is to say that 2*x=4*6. **
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RESPONSE --> understood i used 2x=4(6) is this form ok or would you like the upper star symbol as was used here
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22:19:05 query R.1.50 (was R.1.42) Explain how you evaluate the expression 2 - 5 * 4 - [ 6 * ( 3 - 4) ]
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RESPONSE --> 2 - 5 * 4 - [ 6 * ( 3 - 4) ]=-12 parenthesis first 3-4=-1 6*-1=-6 2-5*4-(-6) 2-20-(-6) =-12
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22:19:13 **Starting with
2-5*4-[6*(3-4)]. First you evaluate the innermost group to get 2-5*4-[6*-1] . Then multiply inside brackets to get 2-5*4+6. Then do the multiplication to get 2-20+6. Then add and subtract in order, obtaining -12. **......!!!!!!!!...................................
RESPONSE --> ok
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22:31:02 query R.1.80 (was R.1.72) Explain how you use the distributive property to remove the parentheses from the express (x-2)(x-4).
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RESPONSE --> (x-2)(x-4)= x(x-4)-2(x-4)= (x^2-4x)+(-2x+8)= x^2-6x+8
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22:31:55 ** COMMON ERROR: Using FOIL. FOIL is not the Distributive Law. FOIL works for binomial expressions. FOIL follows from the distributive law but is of extremely limited usefulness and the instructor does not recommend relying on FOIL.
Starting with (x-2)(x-4) ; one application of the Distributive Property gives you x(x-4) - 2(x-4) . Applying the property to both of the other terms we get x^2 - 4x - (2x -8). Simplifying: x^2 - 4x - 2x + 8 or x^2 - 6x + 8. *......!!!!!!!!...................................
RESPONSE --> understood
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22:39:04 query R.1.86 (was R.1.78) Explain why (4+3) / (2+5) is not equal to 4/2 + 3/5.
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RESPONSE --> you are dividing 2 expressions (4+3)/(2+5) the division bar is acting like 2 sets of parentheses then you would follow the order of operations 1. begin with parentheses 2. multiply & divide left to right 3. perform addition and subtraction left to right
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22:39:36 ** Good answer but at an even more fundamental level it comes down to order of operations.
(4+3)/(2+5) means 7/7 which is equal to 1. By order of operations, in which multiplications and divisions precede additions and subtractions, 4/2+3/5 means (4/2) + (3/5), which gives us 2+3/5 = 2 3/5 **......!!!!!!!!...................................
RESPONSE --> understood
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22:39:49 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> no surprises
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نxWrؒb[ě assignment #002 ؾ{]Uӂ͛ξ College Algebra 09-11-2005
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23:05:58 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= substitute your numbers x=-2,y=3 (2*-2-3)/3= in parentheses do order of operations multiply 1st. (-4-3)/3= -7/3=-2.33...
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23:06:48 ** 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 --> i understand but i took it one step farther and completed division
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23:14:38 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| | substitute x=3,y=-2 | |12| - |-10| |=the |-10| has absolute value around it before we actually do the subtraction in other absolute value marks
|2|=2.................................................
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23:14:47 ** Starting with | | 4x |- | 5y | | we substitute x=3 and y=-2 to get
| | 4*3 | - | 5*-2 | | = | | 12 | - | -10 | | = | 12-10 | = | 2 | = 2. **......!!!!!!!!...................................
RESPONSE --> ok
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23:26:31 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 --> (-9x^2 - x + 1) / (x^3 + x) the denominator cannot =0 it is not defined must be excluded c.x=0
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23:27:00 ** 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. **......!!!!!!!!...................................
RESPONSE --> ok
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23:32:34 query R.2.73 (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 --> (-4)^-2 =16 If a is a real number and n is a positive integer, then the symbol a^n represents the product of n factors of a that is= a^n=a*a
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23:36:50 **Since a^-b = 1 / (a^b), we have
(-4)^-2 = 1 / (-4)^2 = 1 / 16. **......!!!!!!!!...................................
RESPONSE --> i do understand the "whenever you encounter a negative exponent think reciprocal" a^-n =1/a^n if a is not equal to 0 but... -4*-4 is equal to just 16 not 1/16 so could you elaborate please
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23:43:31 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)=If a is a real number and n is a positive integer, then the symbol a^n represents the product of n factors of a that is = a^n=a*a and then you use the parentheses first solving and then multiply and divide from left to right 9*125/9*5=1125/45=25
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23:50:13 ** (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. **......!!!!!!!!...................................
RESPONSE --> i do see in the explanation what had happened
(3^(-2)*5^3)/(3^2*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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23:57:37 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= Whenever you encounter a negative exponent think reciprocala^-n=1/a^n if a is not equal to 0 216y^6/125x^6
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23:59:14 [ 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.......!!!!!!!!...................................
RESPONSE --> i understand i wrote mine out a little farther as in the answer to #93 in review questions in the back of the book answers
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00:04:03 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 --> 1/(-8x^3)^2= whenever you encounter a negative exponent think reciprocal
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00:05:02 ** 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 ). **......!!!!!!!!...................................
RESPONSE --> ok
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00:27:32 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/x^1*y^1-y^2= x^-3*y^-1=x^-3y^-1= 1/x^3y
think reciprocal a-^n=1/a^n if a is not equal to 0 and a^m/a^n=a^m-n= 1/a^n-m.................................................
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00:27:43 ** (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).**......!!!!!!!!...................................
RESPONSE --> ok
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00:40:03 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 ]= 4*25*x^-6*y^-3*z^4= 100z^4*1/x^6y^3= 100z^4/x^6y^3
a^m/a^n=a^m-n=1/a^n-m if a is not equal to zero.................................................
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00:42:09 ** 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 --> 100z^4*1/x^6y^3= 100z^4/x^6y^3 do not understand why 25 was not multiplied by 4 and pulled over like on pg 23 of book example 10 item a
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00:44:03 query R.2.122 (was R.4.72). Express 0.00421 in scientific notation.
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RESPONSE --> 4.21 X 10^-3
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00:44:21 ** 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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00:45:17 query R.2.128 (was R.4.78). Express 9.7 * 10^3 in decimal notation.
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RESPONSE --> 9700
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00:45:22 ** 9.7*10^3 in decimal notation is 9.7 * 1000 = 9700 **
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RESPONSE --> ok
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00:56:11 query R.2.150 (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 --> a. t=97= |t-98.6|<1.6
b.t= 100= | T - 98.6 | _> _ 1.4 b is greater than or equal to.................................................
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00:56:59 ** 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. **......!!!!!!!!...................................
RESPONSE --> understand
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