course Mth 163 Had a few difficulties with this but I was just making things harder than they really were ×ÐòDÝÝ™~¤äˆèòzŠÂx¬÷„ÀæÜassignment #013
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12:18:10 `q001. Note that this assignment has 12 questions What does 2^5 mean?
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RESPONSE --> 2 multiplied by itself 5 times confidence assessment: 3
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12:18:37 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 --> should have given the answer 32 self critique assessment: 2
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12:22:44 `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 --> this means 2 to the third power which is 8 times 2 to the 5th which is 32 so the answer is 256, or it could be expressed as 2^8 confidence assessment: 3
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12:22:53 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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12:23:55 `q003. Why do we say that a^b * a^c = a^(b+c)?
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RESPONSE --> because the problem is simpified to a different form from which you obtain the same number confidence assessment: 3
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12:24:07 03-18-2009 12:24:07 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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NOTES -------> 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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12:26:37 `q004. What does (2^3)^5 mean?
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RESPONSE --> by the order of operations you do what's inside the parethesis first so 2^3 equals 8 and 8 to the fifth equals 32768 confidence assessment: 3
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12:27:27 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 --> 8 to the fifth can be expressed as 2^15 self critique assessment: 2
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12:28:28 `q005. Why do we say that (a^b)^c = a^(b*c)?
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RESPONSE --> because exponents can be multiplied and you get the same answer as you would if you did what was in the parethesis first confidence assessment: 2
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12:28:41 03-18-2009 12:28:41 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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NOTES -------> 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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12:34:44 `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 --> well the answer you get is 8, but the power can't be -10 because you get a completely different answer if you say 2^-10 you get 9.76*10^-4 but when you say (2^5)(2^-2) you get 8 confidence assessment: 1
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12:36:07 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.
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RESPONSE --> I see why I was confused the exponent was not -2*5=-10 it is -2+5 which equals 3 self critique assessment: 2
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12:36:52 `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 --> .25 is the value of 2^-2 confidence assessment: 3
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12:37:13 03-18-2009 12:37:13 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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NOTES -------> 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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12:38:02 `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 exponent has to be simplified confidence assessment: 2
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12:38:14 03-18-2009 12:38:14 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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NOTES -------> 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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12:40:03 `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 --> you get 1 because 3 and -3 cancels out each other and you get zero and anything to the power of 0 is 1 confidence assessment: 3
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12:40:14 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 --> self critique assessment: 3
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12:40:24 `q010. Continuing the last question, what therefore should be the value of 2^0?
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RESPONSE --> 1 confidence assessment:
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12:40:32 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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12:46:00 `q011. How do we solve the equation x^3 = 12?
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RESPONSE --> not sure how to confidence assessment: 0
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12:47:12 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 --> ok I see what to do I thought only negative exponents could be made a fractoin self critique assessment: 2
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12:50:27 `q012. How do we solve the equation x^(5/2) = 44?
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RESPONSE --> this also confuses me I am not sure how to do this confidence assessment: 0
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12:51:31 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 --> making it harder than it is you just do the same thing like in the last problem self critique assessment: 2
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