Assignment 13

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22:22:44 `q001. Note that this assignment has 12 questions What does 2^5 mean?

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RESPONSE --> 2 raised to the 5th power The answer is 32

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22:22:51 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 --> got it

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22:24:33 `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*2*2) * (2*2*2*2*2) 8 * 32 256 which is a power of 2...2^8

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22:24:42 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 --> got it

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22:26:33 `q003. Why do we say that a^b * a^c = a^(b+c)?

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RESPONSE --> When one multiplies a*x (x being any number) by another a^x, the x's can be added therefore a^(b+c) in this example.

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22:27:12 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've got the meaning, I just didn't explain it in the same way.

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22:28:04 `q004. What does (2^3)^5 mean?

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RESPONSE --> (2*2*2)^5...8^5...32768

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22:29:07 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 --> Wow, was I wrong. After examining your answer, one could say that you could multiply the powers to come to the conclusion of 2^15.

Your answer was also correct. 8^5 and 2^15 are the same, both being equal to (2^3)^5.

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22:35:13 `q005. Why do we say that (a^b)^c = a^(b*c)?

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RESPONSE --> You can multiply b8x to find the power that ""a"" would be raised.

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22:35:28 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 --> OK

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22:36:20 `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 --> 2^7...2*(5+2)

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22:37:38 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.

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RESPONSE --> Apparently I didn't look at the question close enough. I didn't see the -2 in the second part of the question. But given it being a -2 it would be 2^(5+ -2) or 2^3

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22:39:14 `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 --> Enter, as appropriate, an answer to the question, a critique of your answer in response to a given answer, your insights regarding the situation at this point, notes to yourself, or just an OK. 2^-2 = .25

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22:40:21 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 did it on the calculator, I didn't even think about doing it in a fraction form. But the calculation stands to reason in the fraction form.

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22:41:26 `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?

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RESPONSE --> The negative symbol in the exponent tells us that the number that is being raised should go to the denominator of the fraction so the power will be positive.

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22:41:40 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 --> OK

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22:42:40 `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 --> 2^0 = 1

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22:42:47 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 --> OK

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22:43:14 `q010. Continuing the last question, what therefore should be the value of 2^0?

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RESPONSE --> 2^0 = 1...in fact, any number raised to the 0 power is always 1

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22:43:22 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 --> ok

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22:46:50 `q011. How do we solve the equation x^3 = 12?

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RESPONSE --> in order to solve this equation, you would have to find the cubed root of each side which would say: I don't know how to type this but the ""jist"" is cuber root of x^3 = cubed root of 12

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22:48:31 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 --> you are so correct...I was trying to make it harder than it really is...by multiplying both sides by the reciprical of the power of 3 (1/3) you would eliminate the x^3 to x as long as the 12 was divided by the same 1/3.

Your use of the cube root is perfectly appropriate. The cube root is the 1/3 power, and the 1/3 power is the cube root.

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22:50:28 `q012. How do we solve the equation x^(5/2) = 44?

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RESPONSE --> x^5/2 / (2/5) = 44 / (2/5)

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22:51:50 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 --> Again, I goofed. I divided both sides by the 2/5 instead of multiplying the power by its reciprical. Doing so would solve the problem and my answer matches the one given

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Good. Let me know if you have questions.