course Mth 163 8-2-09 at 2:00 013.
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2 to the 3rd power times 2 to the 5th poser 2^3x2^5 8x32 256 2^8 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q003. Why do we say that a^b * a^c = a^(b+c)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When you raise anything (a) by exponents of the same base (b and c) you add the exponents confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. What does (2^3)^5 mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (2^3)*(2^3)*(2^3)*(2^3)*(2^3)=2^15 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q005. Why do we say that (a^b)^c = a^(b*c)? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: a^b stands for a multiplied by itself b times. (a^b)^c is a^b multiplied by itself c times multiplying a by itself b * c times. (a^b)^c = a^(b * c). confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: , 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^5 = 32 and 2^3 = 8 32 * 2^-2 = 8 2^-2 = 8 / 32 = 1/4. confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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 why 2^-2 = 1/2^2. QUESTIONABLE STUDENT RESPONSE: .25 .25 is of course the value of 2^-2. However I'm not sure you've connected this with the fact that 2^5 * 2^-2 = 2^3, as was the intent of the question. A key word in the given question is 'therefore', which asks you to connect your answer to the fact that 2^5 * 2^-2 = 2^3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The negative sign in front of the “n” means to do the opposite of multiplying the number which is dividing. confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2^3 * 2^-3 = 2^(3 + -3) = 2^(3-3) = 2^0. 2^-3 = 1 / 2^3 2^3 * 2^-3 = 2^3 * ( 1 / 2^3) = 1. confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q010. Continuing the last question, what therefore should be the value of 2^0? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Any number raised to 0 equals 1 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q011. How do we solve the equation x^3 = 12? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: You take the cube root of each side and the cube root of 12 is 2.29 approximately confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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 . &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q012. How do we solve the equation x^(5/2) = 44? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^(5/2 * 2/5) = 44^(2/5) x^1 = 44^(2/5) x = 44^(2/5) X=4.54 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: 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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: