course mth 163 \????E?????????assignment #012012.
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19:34:41 `q001. Note that this assignment has 3 questions If we know that y = k x^2, then if (x2/x1) = 7, what is (y2/y1)?
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RESPONSE --> If (x2/x1) = 7 then x2 = 7x1 and x1 = (x2/7) When you substitute in to get y2 and y1: y2 = k(7x1)^2 y2 = k49(x1^2) y1 = k(x2/7)^2 y1 = (k(x2^2))/49 Then put that into y2/y1 [k49(x1^2)] / (k(x2^2))/49 - multiply by the reciprical [k49(x1^2)]*[49/k(x2^2)] (2401(x1^2)k) / kx2^2 - k is cancelled out (2401x1^2) / x2^2 confidence assessment: 1
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19:34:47 `q001. Note that this assignment has 3 questions If we know that y = k x^2, then if (x2/x1) = 7, what is (y2/y1)?
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RESPONSE --> confidence assessment:
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19:36:18 If y2 = k x2^2 and y1 = k x1^2, then y2 / y1 = (k x2^2) / ( k x1^2). Since k / k = 1 this is the same as y2 / y1 = x2^2 / x1^2, which is the same as y2 / y1 = (x2 / x1)^2. In words this tells us if y to is proportional to the square of x, then the ratio of y2 to y1 is the same as the square of the ratio of x2 to x1. Now if (x2 / x1) = 7, we see that y2 / y1 = (x2 / x1)^2 = 7^2 = 49.
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RESPONSE --> I made this problem harder than it was. I do understand how the answer was gotten. self critique assessment: 3
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19:43:59 `q002. If we know that y = k x^3, then if (x2/x1) = 7, what is (y2/y1)?
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RESPONSE --> From the previous question we know that y2/y1 is equal to (x2/x1)^3 Since we know that x2/x1 = 7, then (x2/x1)^3 would be 7^3 = 343 confidence assessment: 3
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19:44:38 If y2 = k x2^3 and y1 = k x1^3, then y2 / y1 = (k x2^3) / ( k x1^3). Since k / k = 1 this is the same as y2 / y1 = x2^3 / x1^3, which is the same as y2 / y1 = (x2 / x1)^3. In words this tells us if y to is proportional to the cube of x, then the ratio of y2 to y1 is the same as the cube of the ratio of x2 to x1. Now if (x2 / x1) = 7, we see that y2 / y1 = (x2 / x1)^3 = 7^3 = 343.
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RESPONSE --> self critique assessment: 3
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19:51:52 `q003. If we know that y = k x^-2, then if (x2/x1) = 64, what is (y2/y1)?
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RESPONSE --> Again, from previous questions: we find that y2 = k x2^-2 and y1 = k x1^-2 This means that y2 / y1 = (k x2^-2) / ( k x1^-2) and k is canceled out leaving y2 / y1 = (x2 / x1)^-2 Since (x2 / x1) = 64, then y2 / y1 = (x2 / x1)^-2 = 64^-2 = .00024 confidence assessment: 2
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19:52:16 If y2 = k x2^-2 and y1 = k x1^-2, then y2 / y1 = (k x2^-2) / ( k x1^-2). Since k / k = 1 this is the same as y2 / y1 = x2^-2 / x1^-2, which is the same as y2 / y1 = (x2 / x1)^-2, which is the same as 1 / (x2 / x1)^2, which gives us (x1 / x2)^2. So if y = k x^-2, then (y2 / y1) = (x1 / x2)^2.( In words this tells us if y to is inversely proportional to the square of x, then the ratio of y2 to y1 is the same as the square of the ratio of x1 to x2 (note that this is a reciprocal ratio). Now if (x2 / x1) = 64, we see that y2 / y1 = (x1 / x2)^2 = (1/64)^2 = 1/ 4096.
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RESPONSE --> the decimal I put down is equivalient to 1/4096 self critique assessment: 3
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??????????? assignment #013 013. Precalculus I 02-26-2007
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19:55:33 `q001. Note that this assignment has 12 questions What does 2^5 mean?
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RESPONSE --> 2^5 means 2 raised to the fifth power or 2*2*2*2*2 confidence assessment: 3
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19:56:35 2^5 stands for 2 raised to the fifth power; i.e., 2^5 = 2*2*2*2*2. The result of this calculation is 2^5 = 32.
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RESPONSE --> I did not put down the final conclusion self critique assessment: 2
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20:02:17 `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it?
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RESPONSE --> 2^3 * 2^5 means (2*2*2) * (2*2*2*2*2) which turns out to be 8 * 32, which equals 256. 256 is a power of 2 - it is 2^8 Using the law of exponents 2^3 * 2^5 is the same as 2^(3+5) or 2^8 confidence assessment: 3
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20:02:32 2^3 * 2^5 means (2*2*2) * (2*2*2*2*2). This is the same as 2*2*2*2*2*2*2*2, or 2^8. When we multiply this number out, we obtain 256.
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RESPONSE --> self critique assessment: 3
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20:03:56 `q003. Why do we say that a^b * a^c = a^(b+c)?
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RESPONSE --> It is due to the Law of Exponents. confidence assessment: 2
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20:07:25 We saw in the preceding example that 2^3 * 2^5 stood for a product of three 2's, multiply by a product of five 2's. We saw also that the result was identical to a product of eight 2's. This was one instance of the general rule that when we multiply to different powers of the same number, the result is that number raised to the sum of the two powers. One general way to state this rule is to let a stand for the number that is being raised to the different powers, and let b and c stand for those powers. Then we get the statement a^b * a^c = a^(b+c).
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RESPONSE --> I didn't even think to use the previous question as an example, but I understand as long as you are using the same number in this case 2, to be raised to a power, then you can add exponents. self critique assessment: 2
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20:10:30 `q004. What does (2^3)^5 mean?
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RESPONSE --> (2^3)^5 means (2*2*2)^5 or 8*8*8*8*8 or32768 confidence assessment: 3
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20:11:09 Since 2^3 stands for 2*2*2, it follows that (2^3)^5 means (2^3)*(2^3)*(2^3)*(2^3)*(2^3) = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)*(2*2*2) = 2*2*2*2*2*2*2*2*2*2*2*2*2*2*2 = 2^15.
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RESPONSE --> self critique assessment: 3
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20:24:25 `q005. Why do we say that (a^b)^c = a^(b*c)?
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RESPONSE --> In the last question, (2^3)^5 stood for 2^3 raised to the fith power or the product of five sets of (2*2*2). It was also identical to the product fifteen 2's. In this example a rule can be seen that when you raise something that has been previously raised to another power, the result is the same as if you multiplied the two powers. confidence assessment: 2
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20:24:51 We saw in the last example how (2^3)^5 stands for the product of 5 quantities, each equal to the product of three 2's. We saw how this is equivalent to the product of fifteen 2's, and we saw how the fifteen was obtained by multiplying the exponents 3 and 5. In the present question a^b stands for the quantity a multiplied by itself b times. (a^b)^c stands for the quantity a^b multiplied by itself c times, which is equivalent to multiplying a by itself b * c times. Thus we say that (a^b)^c = a^(b * c).
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RESPONSE --> This says it much better than I did, but I think I said the same thing. self critique assessment: 3
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20:29:03 `q006. According to the law a^b * a^c = a*(b+c), if we multiply 2^5 by 2^-2 what power of 2 should we get?
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RESPONSE --> you should get 2^3 confidence assessment: 2
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20:33:44 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.
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RESPONSE --> I just didn't show the full process self critique assessment: 2
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20:35:52 `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2?
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RESPONSE --> the value of 2^-2 is the same as (1/2)*(1/2) which is 1/4 or .25 confidence assessment: 2
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20:41:12 One way of seeing this is to note that 2^5 = 32 and 2^3 = 8, so we have 32 * 2^-2 = 8. Dividing both sides by 32 we get 2^-2 = 8 / 32 = 1/4. We can learn something important if we keep the calculation in powers of 2. If 2^5 * 2^-2 = 2^3, then dividing both sides of the equation by 2^5 we obtain 2^-2 = 2^3/2^5, which is equal to 1/2^2. This shows us that 2^-2 = 1/2^2.
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RESPONSE --> I wasn't sure of a good way to put this, but understand the concept self critique assessment: 3
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20:44:35 `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?
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RESPONSE --> because a negative power make the number the inverse. confidence assessment: 1
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20:45:34 This is because for any number m, we have 2^m * 2^-n = 2^( m + -n) = 2^(m-n), and we also have 2^m * (1 / 2^n) = 2^m / 2^n = 2^(m-n). So whether we multiply 2^m by 2^-n or by 1 / 2^n we get the same result. This shows that 2^-n and 1 / 2^n are the same.
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RESPONSE --> I understand the concept, just was not able to put it quite in the terms the explaination did. self critique assessment: 2
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20:47:32 `q009. According to the law a^b * a^c = a*(b+c), if we multiply 2^3 by 2^-3 what power of 2 should we get? Since 2^-3 = 1 / 2^3, what number must we get when we multiply 2^3 by 2^-3?
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RESPONSE --> The power you would get is: 2^(3+-3) or 2^0 which equals 1 confidence assessment: 2
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20:48:12 2^3 * 2^-3 = 2^(3 + -3) = 2^(3-3) = 2^0. Since 2^-3 = 1 / 2^3 it follows that 2^3 * 2^-3 = 2^3 * ( 1 / 2^3) = 1.
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RESPONSE --> I thought the previous explaination that I gave sufficed for both, but I was off just a little. self critique assessment: 3
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20:48:31 `q010. Continuing the last question, what therefore should be the value of 2^0?
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RESPONSE --> the value of 2^0 is equal to 1 confidence assessment: 3
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20:48:39 Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1.
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RESPONSE --> self critique assessment: 3
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20:58:00 `q011. How do we solve the equation x^3 = 12?
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RESPONSE --> You take the cubed root of 12 confidence assessment: 1
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20:59:15 We solve the equation by taking the 1/3 power of both sides: (x^3)^(1/3) = 12^(1/3), then by the law (a^b)^c = a^(bc) we have x^(3 * 1/3) = 12^(1/3), so that x^1 = 12^(1/3) or just x = 12^(1/3), which we can easily enough evaluate with a calculator. We obtain a result of approximately x = 2.29 .
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RESPONSE --> This makes sense...I am more familiar with the cubed root function of the calculator. self critique assessment: 2
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21:06:17 `q012. How do we solve the equation x^(5/2) = 44?
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RESPONSE --> We would raise both sides by 2/5 power: x^(5/2)^2/5 = 44^2/5 Using the law (a^b)^c = a^(bc) we can make: x^(5/2 * 2/5) = 44^(2/5) = x^1 = 44^(2/5) or x = 44^(2/5) which equals 4.54 confidence assessment: 3
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21:06:22 If we take the 2/5 power of both sides we get (x^(5/2))^(2/5) = 44^(2/5) or x^(5/2 * 2/5) = 44^(2/5) or x^1 = 44^(2/5) so that x = 44^(2/5). Evaluating by calculator you should obtain approximately x = 4.54.
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RESPONSE --> self critique assessment: 3
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