course MTH 163 I'm not sure if you received these two assignments or not...please excuse them if you did; i am sorry.
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21:54:14 `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 (y2/y1) will equal k49 ??? I'm not sure how to solve the problem.
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21:56:16 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. 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 --> Well oddly I got the numerical part of my answer. I just shouldn't have put k infront of it. I'm understanding how to solve them now.
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21:58:29 `q002. If we know that y = k x^3, then if (x2/x1) = 7, what is (y2/y1)?
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RESPONSE --> If (x1/x1) = 7 when y = k x^3, then (y2/y1) = 343. ??
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21:59:04 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 --> i was correct.
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22:00:21 `q003. If we know that y = k x^-2, then if (x2/x1) = 64, what is (y2/y1)?
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RESPONSE --> If (x2/x1) = 64 when y = k x^2, then (y2/y1) = 4096. ??
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22:02:13 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 --> I was almost correct....I put 4096 instead of 1/4096.
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UK撰ؾޗ掠 Student Name: assignment #013
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22:03:40 `q001. Note that this assignment has 12 questions What does 2^5 mean?
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RESPONSE --> 2^5 means 2 to the 5th power or 2*2*2*2*2 which equals 32.
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22:04:07 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 was correct.
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22:08:30 `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 that 2 raised to the 3rd power multiplied by 2 raised to the 5th power. Or it means (2*2*2) * (2*2*2*2*2) = (8) * (32) = 256 Yes, the result is a power of two. It is 16^2.
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22:09:00 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 --> Yes, I was correct.
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22:11:24 `q003. Why do we say that a^b * a^c = a^(b+c)?
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RESPONSE --> Because when we simplify a^b * a^c, then we get a^(b+c). Since a + a is a, then the remaining variables (b and c) will become (b + c). ???
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22:12:44 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'm not sure if my answer was similar to yours, but i understand what you're saying.
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22:15:35 `q004. What does (2^3)^5 mean?
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RESPONSE --> (2^3)^5) means that the product of 2 raised to the 3rd power is also raised to the 5th power. (2*2*2) * (2*2*2) * (2*2*2) * (2*2*2) * (2*2*2) Or 2^3 = 8 = 8^5 = 32768
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22:16:35 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 --> I was correct, except I approached it a different way and i went ahead and solved it.
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22:19:50 `q005. Why do we say that (a^b)^c = a^(b*c)?
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RESPONSE --> Because it's the same as a^b * a^c = a^(b+c) and the concept in the previous question. You will still get the same answer by writing it either way.
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22:24:45 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 --> I'm not sure if i was correct or not, but I think i understand what you're saying.
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22:26:37 `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 --> We should get 2^(5+2) or 2^7.
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22:27:56 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.
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RESPONSE --> I overlooked that little symbol between the two 2's. I thought it was a regular ""raised power"" symbol. I understand.
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22:30:34 `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 must be 2^ -2 ??
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22:32:23 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 didn't get the correct answer. i don't quite understand the problem and the concept of 2^-2.
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22:35:03 `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?
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RESPONSE --> Because 2^-2 = 1/ 2^2 therfore you can change the value of n, but it'll be 1/ 2^n. I'm not sure If i understand what is being said/asked in this question; that Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?
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22:36:29 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 was correct in saying that 2^-n is the same as 1 / 2^n, but I don't really understand the first part of the answer: 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.
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22:39:32 `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 --> According to the law, 2^3 * 2^-3 = 2^(3 + -3) = 2^(0) = 1. ???
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22:40:04 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 believe i was correct.
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22:40:41 `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 1.
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22:40:59 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 --> I was correct.
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22:44:48 `q011. How do we solve the equation x^3 = 12?
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RESPONSE --> I'm not sure how to solve the equation x^3 = 12 ???
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22:45:57 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 --> I understand now.
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22:51:15 `q012. How do we solve the equation x^(5/2) = 44?
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RESPONSE --> We solve the equation by taking the (1/ (5/2) power of both sides: (x^5/2)^(1/(5/2) = 44^(1/(5/2), then by the law of (a^b)^c = a^(bc) we have x^(5/2 * 1/(5/2) = 44^(1/(5/2)). We obtain a result of 4.54. ???
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22:52:53 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 --> I got the same answer but I didn't take the 2/5 power from each side. I took the 1/(5/2) power.
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