course Mth 163 xEۖyassignment #012
......!!!!!!!!...................................
15:31:56 problem 1. box of length 30 centimeters capacity 50 liters . What is the proportionality for this situation, what is the proportionality constant and what is the specific equation that relates capacity y to length x?
......!!!!!!!!...................................
RESPONSE --> The proportionality for this situation is y2/y1 = (x2/x1)^3 (30,50); 50=k(30)^3 => 50=24000k => k=1/540 or .0019 so the specific equation is y=(1/540)x^3
.................................................
......!!!!!!!!...................................
15:32:07 ** The proportionality for volume is y = k x^3, where y is capacity in liters when x is length in cm. Since y = 50 when x = 30 we have 50 = k * 30^3 so that k = 50 / (30^3) = 50 / 27,000 = 1/540 = .0019 approx. Thus y = (1/540) * x^3. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:33:36 What is the storage capacity of a box of length 100 centimeters?
......!!!!!!!!...................................
RESPONSE --> (?cm, 100 liters) y=(1/540)x^3 100=(1/540)x^3, divide both sides by 1/540 x^3 = 54000, raise both sides to 1/3 x=37.798 cm
.................................................
......!!!!!!!!...................................
15:36:01 ** The proportionality is y = 1/540 * x^3 so if x = 100 we have y = 1/540 * 100^3 = 1900 approx. A 100 cm box geometrically similar to the first will therefore contain about 1900 liters. **
......!!!!!!!!...................................
RESPONSE --> I typed in the problem previous to this one on the page What I did for (100cm,?liters) is as follows: y=(1/540)x^3 y=(1/540)(100)^3 y=1851.852 liters.
.................................................
......!!!!!!!!...................................
15:36:27 What length is required to obtain a storage capacity of 100 liters?
......!!!!!!!!...................................
RESPONSE --> (?cm, 100 liters) y=(1/540)x^3 100=(1/540)x^3, divide both sides by 1/540 x^3 = 54000, raise both sides to 1/3 x=37.798 cm
.................................................
......!!!!!!!!...................................
15:37:26 ** If y = 100 then we have 100 = (1/540) * x^3 so that x^3 = 540 * 100 = 54,000. Thus x = (54,000)^(1/3) = 185 approx. The length of a box that will store 100 liters is thus about 185 cm. **
......!!!!!!!!...................................
RESPONSE --> 54000^(1/3) is approx 37.798
.................................................
......!!!!!!!!...................................
15:41:12 How long would a box have to be in order to store all the water in a swimming pool which contains 450 metric tons of water? A metric ton contains 1000 liters of water.
......!!!!!!!!...................................
RESPONSE --> 450 metric tons * 1000 liters = 450000 liters (?cm, 450000 liters) y=(1/540)x^3 450000 = (1/540)x^3, divide both sides by 1/540 243000000 = x^3, raise both sides to 1/3 power x= 624.03 cm
.................................................
......!!!!!!!!...................................
15:41:17 ** 450 metric tons is 450 * 1000 liters = 450,000 liters. Thus y = 450,000 so we have the equation 540,000 = (1/540) x^3 which we solve in a manner similar to the preceding question to obtain x = 624, so that the length of the box is 624 cm. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:42:30 problem 2. cleaning service scrub the surface of the Statute of width of finger .8 centimeter vs. 20-centimeter width actual model takes .74 hours. How long will it take to scrub the entire statue?
......!!!!!!!!...................................
RESPONSE --> (20cm,?hrs) y=1.15625(20)^2 y=1.15625(400) y=462.5 hours
.................................................
......!!!!!!!!...................................
15:42:37 ** y = k x^2 so .74 = k * .8^2. Solving for k we obtain k = 1.16 approx. so y = 1.16 x^2. The time to scrub the actual statue will be y = 1.16 x^2 with x = 20. We get y = 1.16 * 20^2 = 460 approx.. It should take 460 hrs to scrub the entire statue. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:44:25 problem 3. illumination 30 meters is 5 foot-candles. What is the proportionality for this situation, what is the value of the proportionality constant and what equation relates the illumination y to the distance x?
......!!!!!!!!...................................
RESPONSE --> The proportionality is y=kx^(-2) (30m,50ft candle); 5=k(30)^(-2) => k=4500 so the resulting equation is y=4500x^(-2)
.................................................
......!!!!!!!!...................................
15:46:21 ** The proportionality should be y = k x^-2, where y is illumination in ft candles and x the distance in meters. We get 5 = k * 30^-2, or 5 = k / 30^2 so that k = 5 * 30^2 = 4500. Thus y = 4500 x^-2. We get an illumination of 10 ft candles when y = 10. To find x we solve the equation 10 = 4500 / x^2. Multiplying both sides by x^2 we get 10 x^2 = 4500. Dividing both sides by 10 we have x^2 = 4500 / 10 = 450 and x = sqrt(450) = 21 approx.. For illumination 1000 ft candles we solve 1000 = 4500 / x^2, obtaining solution x = 2.1 approx.. We therefore conclude that the comfortable range is from about x = 2.1 meters to x = 21 meters. **
......!!!!!!!!...................................
RESPONSE --> The question didn't seem like it asked for the problem of (?m,10 ft candles) but I did not have a problem working that out to arrive at the same answer, the same goes for (?m, 1000ft candles)
.................................................
......!!!!!!!!...................................
15:47:23 problem 5. Does a 3-unit cube weigh more or less than 3 times a 1-unit cube? Why is this?
......!!!!!!!!...................................
RESPONSE --> More because if you just multiplied 35 * 3 to find the weight of the 3-unit you aren't taking into consideration the depth change between the two cubes along with the height and width changes.
.................................................
......!!!!!!!!...................................
15:47:36 ** A 3-unit cube is equivalent to 3 layers of 1-unit cubes, each layer consisting of three rows with 3 cubes in each row. Thus a 3-unit cube is equivalent to 27 1-unit cubes. If the weight of a 1-unit cube is 35 lbs then we have the following: Edge equiv. # of weight Length 1-unit cubes 1 1 35 2 4 4 * 35 = 140 3 9 9 * 35 = 315 4 16 16 * 35 = 560 5 25 25 * 35 = 875. Each weight is obtained by multiplying the equivalent number of 1-unit cubes by the 35-lb weight of such a cube. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:49:53 problem 6. Give the numbers of 1-unit squares required to cover 6-, 7-, 8-, 9- and 10-unit square, and also an n-unit square.
......!!!!!!!!...................................
RESPONSE --> 6^2 = 36 1-unit squares 7^2 = 49 1-unit squares 8^2 = 64 1-unit squares 9^2 = 81 1-unit squares 10^2 = 100 1-unit squares n^2
.................................................
......!!!!!!!!...................................
15:50:03 ** To cover a 6-unit square requires 6 rows each containing 6 1-unit squares for a total of 36 one-unit squares. To cover a 7-unit square requires 7 rows each containing 7 1-unit squares for a total of 49 one-unit squares. To cover a 8-unit square requires 8 rows each containing 8 1-unit squares for a total of 64 one-unit squares. To cover a 9-unit square requires 9 rows each containing 9 1-unit squares for a total of 81 one-unit squares. To cover a 10-unit square requires 10 rows each containing 10 1-unit squares for a total of 100 one-unit squares. To cover an n-unit square requires n rows each containing n 1-unit squares for a total of n*n=n^2 one-unit squares. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:50:10 problem 8. Relating volume ratio to ratio of edges.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:53:38 ** right idea but you have the ratio upside down. The volume ratio of a 5-unit cube to a 3-unit cube is (5/3)^3 = 125 / 27 = 4.7 approx.. The edge ratio is 5/3 = 1.67 approx. VOlume ratio = edgeRatio^3 = 1.678^3 = 4.7 approx.. From this example we see how volume ratio = edgeRatio^3. If two cubes have edges 12.7 and 2.3 then their edge ratio is 12.7 / 2.3 = 5.5 approx.. The corresponding volume ratio would therefore be 5.5^3 = 160 approx.. If edges are x1 and x2 then edgeRatio = x2 / x1. This results in volume ratio volRatio = edgeRatioo^3 = (x2 / x1)^3. **
......!!!!!!!!...................................
RESPONSE --> I arrived at the same conclusion that VolRation = edgeRatio^3
.................................................
......!!!!!!!!...................................
15:54:49 problem 9. Relating y and x ratios for a cubic proportionality. What is the y value corresponding to x = 3 and what is is the y value corresponding to x = 5?
......!!!!!!!!...................................
RESPONSE --> y1 = a(3)^3 = 27a y2 = a(5)^3 = 125a
.................................................
......!!!!!!!!...................................
15:55:09 ** If y = a x^3 then if x1 = 3 we have y1 = a * 3^3 and if x2 = 5 we have y2 = a * 5^3. This gives us ratio y2 / y1 = (a * 5^3) / (a * 3^3) = (a / a) * (5^3 / 3^3) = 1 * 125 / 27 = 125 / 27. In general if y1 = a * x1^3 and y2 = a * x2^3 we have } y2 / y1 = (a x2^3) / (a x1^3) = (a / a) * (x2^3 / x1^3) = (x2/x1)^3. This tells you that to get the ratio of y values you just cube the ratio of the x values. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:55:24 problem 10. Generalizing to y = x^p.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:59:26 ** If y = a x^2 then y2 / y1 = (a x2^2) / (a x1^2) = (a / a) * (x2^2 / x1^2) = (x2/x1)^2. This tells you that to get the ratio of y values you just square the ratio of the x values. If y = f(x) = a x^p then y1 = f(x1) = a x1^p and y2 = f(x1) = a x2^p so that y2 / y1 = f(x2) / f(x1) = (a x2^p) / (a x1^p) = (a / a) ( x2^p / x1^p ) = x2^p / x1^p = (x2 / x1)^p. **
......!!!!!!!!...................................
RESPONSE --> When the previous statement doesn't ask a question it makes me think that it is just separating the sections, instead it wants you to enter what you got for that question. However, I arrived at the exact same conclusion that y2/y1 = (x2/x1)^2 so y2/y1 = (x2/x1)^p generally.
.................................................
......!!!!!!!!...................................
15:59:30 Add comments on any surprises or insights you experienced as a result of this assignment.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:59:35 21:40:33
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
15:59:43 this was a pretty easy assignment to comprehend, I did like the ratio stuff looks like it will come in handy ** this stuff is very important in most areas of study **
......!!!!!!!!...................................
RESPONSE -->
.................................................
ˊG assignment #013 ҶQ̫ Precalculus I 10-10-2006
......!!!!!!!!...................................
23:08:19 Query last asst before test 1, problem 1.Give your solution to x ^ 3 / 17 = 58
......!!!!!!!!...................................
RESPONSE --> x ^ 3 / 17 = 58; multiply both sides by 17 x^3 = 986; raise both sides to the 1/3 power x=986^1/3 x = approx. 9.95
.................................................
......!!!!!!!!...................................
23:08:30 ** The solution to x^3 / 17 = 58 is found by first multiplying both sides by 17 to get x^3 = 58 * 17 then taking the 1/3 power of both sides, obtaining (x^3)^(1/3) = (58 * 17)^(1/3) or x = 9.95, approx.. COMMON ERROR: If you interpret the equation as x^(3/17) = 58 you will get solution x = 58^(17/3) = 9834643694. However this is not the solution to the given equation To interpret x ^ 3 / 17 you have to follow the order of operations. This means that x is first cubed (exponentiation precedes multiplication or division) then divided by 17. If you introduce the grouping x^(3/17) you are changing the meaning of the expression, causing 3 to be divided by 17 before exponentiation. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:13:25 Give your solution to (3 x) ^ -2 = 19
......!!!!!!!!...................................
RESPONSE --> (3 x) ^ -2 = 19; raise both sides to the -1/2 power 3x=19^(-1/2); evaluate 19^(-1/2) and divide by 3 x=.0765
.................................................
......!!!!!!!!...................................
23:13:36 ** (3x)^-2 = 19 is solved by taking the -1/2 power of both sides, or the negative of the result: ((3x)^-2)^(-1/2)) = 19^(-1/2) gives us 3x = 19^(-1/2) so that x = [ 19^(-1/2) ] / 3 = .0765 or -.0765. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:14:54 Give your solution to 4 x ^ -.5 = 7
......!!!!!!!!...................................
RESPONSE --> 4 x ^ -.5 = 7; raise both sides to the -2 power 4x = 7^-2; evaluate 7^-2 and divide by 4 x=.0051
.................................................
......!!!!!!!!...................................
23:16:31 ** to solve the equation we first multiply both sides by 1/4 to get x ^ -.5 = 7 / 4. Then we raise both sides to the -2 power: (x^-.5)^-2 = (7/4)^-2 so x = .327 approx **
......!!!!!!!!...................................
RESPONSE --> in 4 x ^ -.5 = 7 the 4 isn't raise to the -.5 power along with the x, therefore it needs to be raised to the -2 along with the 7.
.................................................
......!!!!!!!!...................................
23:18:30 Give your solution to 14 x ^ (2/3) = 39
......!!!!!!!!...................................
RESPONSE --> 14 x ^ (2/3) = 39; divide both sides by 14 x^(2/3) = 30/14; raise both sides to power 3/2 x=(30/14)^(3/2) x=4.649
.................................................
......!!!!!!!!...................................
23:18:33 ** We first multiply both sides by 1/14 to get x^(2/3) = 39/14. The we raise both sides to the 3/2 power to get x = (39/14)^(3/2) = 4.65. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:26:12 Give your solution to 5 ( 3 x / 8) ^ (-3/2) = 9
......!!!!!!!!...................................
RESPONSE --> 5 ( 3 x / 8) ^ (-3/2) = 9; divide both sides by 5 ( 3 x / 8) ^ (-3/2) = 9/5; raise both sides to the -2/3 3x/8 = (9/5)^(-2/3); multiply by 8 and divide by 3 x=1.802
.................................................
......!!!!!!!!...................................
23:26:15 ** multiplying both sides by 1/5 we get (3x/8)^(-3/2) = 9/5. Raising both sides to the -2/3 power we have 3x / 8 = (9/5)^(-2/3). Multiplying both sides by 8/3 we obtain x = 8/3 * (9/5)^(-2/3) = 1.80 **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:33:49 Query problem 2. a(n+1) = a(n) + .5 n, a(0) = 2 What are a(1), a(2), a(3), a(4) and a(5)?
......!!!!!!!!...................................
RESPONSE --> a(n+1) = a(n) + .5 n, a(0) = 2 a(0+1) = 2 + .5(0), a(1) = 2 a(1+1) = 2 + .5(1), a(2) = 2.5 a(2+1) = 2.5 + .5(2), a(3) = 3.5 a(3+1) = 3.5 + .5(3), a(4) = 5 a(4+1) = 5 + .5(4), a(5) = 10
.................................................
......!!!!!!!!...................................
23:35:12 ** Substituting n = 0 we get a(0+1) = a(0) + .5 * 0 which we simplify to get a(1) = a(0). Substituting a(0) = 2 from the given information we get a(1) = 2. Substituting n = 1 we get a(1+1) = a(1) + .5 * 1 which we simplify to get a(2) = a(1) + .5. Substituting a(1) = 2 from the previous step we get a(2) = 2.5. Substituting n = 2 we get a(2+1) = a(2) + .5 * 2 which we simplify to get a(3) = a(2) + 1. Substituting a(2) = 2.5 from the previous step we get a(3) = 2.5 + 1 = 3.5. Substituting n = 3 we get a(3+1) = a(3) + .5 * 3 which we simplify to get a(4) = a(3) + 1.5. Substituting a(3) = 3.5 from the previous step we get a(4) = 3.5 + 1.5 = 5. Substituting n = 4 we get a(4+1) = a(4) + .5 * 4 which we simplify to get a(5) = a(4) + 2. Substituting a(4) = 5 from the previous step we get a(5) = 5 + 2 = 7. **
......!!!!!!!!...................................
RESPONSE --> on the last one a(5) i accidently multiplied 5*.5(4) instead of adding, that is why i arrived at the answer of 10 instead of 7
.................................................
......!!!!!!!!...................................
23:42:59 What is your quadratic function and what is its value for n = 4? Does it fit the sequence exactly?
......!!!!!!!!...................................
RESPONSE --> I am not sure how to go from a distance equation of a(n+1) = a(n) + .5n to a quadratic function.
.................................................
......!!!!!!!!...................................
23:50:50 ** Using points (1,2), (3,3.5) and (7,5) we substitute into the form y = a x^2 + b x + c to obtain the three equations 2 = a * 1^2 + b * 1 + c 3.5 = a * 3^2 + b * 3 + c 7 = a * 5^2 + b * 5 + c. Solving the resulting system for a, b and c we obtain a = .25, b = -.25 and c = 2, giving us the equation 0.25x^2 - 0.25x + 2. **
......!!!!!!!!...................................
RESPONSE --> Of course you would use a system of equations, that makes sense because you can use three of the points to make three equations to form a system of equations to solve in order to find the values of the three unknowns.
.................................................
......!!!!!!!!...................................
23:56:15 Query problem 3. f(x) = .3 x^2 - 4x + 7, evaluate at x = 0, .4, .8, 1.2, 1.6 and 2.0.
......!!!!!!!!...................................
RESPONSE --> (0, 7) (.4, 5.448) (.8, 3.992) (1.2, 2.632) (1.6, 1.368) (2.0, 0.2)
.................................................
......!!!!!!!!...................................
23:57:02 ** We obtain the points (0, 7) (4, 5.448) (.8, 3.992) (1.2, 2.632) (1.6, 1.368) (2, .2) y values are 7, 5.448, 3.992, 2.632, 1.368, 0.2. Differences are 7-5.448 = -1.552, 3.992 - 2.632 = -1.456, etc. The sequence of differences is -1.552, -1.456, -1.36, -1.264, -1.168. The rate of change of the original sequence is proportional to this sequence of differences. The differences of the sequence of differences (i.e., the second differences) are .096, .096, .096, .096, .096.. These differences are constant, meaning that the sequence of differences is linear.. This constant sequence is proportional to the rate of change of the sequence of differences. The differences are associated with the midpoints of the intervals over which they occur. Therefore the difference -1.552, which occurs between x = 0 and x = .4, is associated with x = .2; the difference -1.456 occuring between x = .4 and x = .8 is associated with x = .6, etc.. The table of differences vs. midpoints is } 0.2, -1.552 -.6, -1.456 1, -1.36 1.4, -1.264 1.8, -1.168 This table yields a graph whose slope is easily found to be constant at .24, with y intercept -1.6. The function that models these differences is therefore y = 2.4 x - 1.6. **
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:58:00 Query problem 4. f(x) = a x^2 + b x + c What symbolic expression stands for the average slope between x = h and x = k?
......!!!!!!!!...................................
RESPONSE --> y=a(k-h)+b
.................................................
......!!!!!!!!...................................
23:58:31 10-10-2006 23:58:31 ** The average slope is rise / run = [ f(k) - f(h) ] / (k - h) = ( a k^2 + b k + c - ( a h^2 + b h + c) ) / ( k - h). We simplify this to get ave slope = ( a ( k^2 - h^2) + b ( k - h) ) / ( k - h), which we write as ave slope = ( a ( k-h) ( k+h) ) + b ( k - h) ) / (k - h). k - h is a factor of the numerator so we have the final form ave slope = a ( k + h) + b. **
......!!!!!!!!...................................
NOTES -------> a(k+h)+b
.................................................
ŃVǖ}OdF] Student Name: assignment #013
.................................................
......!!!!!!!!...................................
22:47:37 `q001. Note that this assignment has 12 questions What does 2^5 mean?
......!!!!!!!!...................................
RESPONSE --> 2^5 means two raised to the fifth power or 2 multiplied by itself 5 times 2*2*2*2*2 = 32
.................................................
......!!!!!!!!...................................
22:47:43 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:49:48 `q002. What does 2^3 * 2^5 mean? Is the result of power of 2? If so, what power of 2 is it?
......!!!!!!!!...................................
RESPONSE --> 2^3*2^5 means (2*2*2)(2*2*2*2*2) = 8*32 = 256, which is equal to 2^8
.................................................
......!!!!!!!!...................................
22:49:54 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:51:56 `q003. Why do we say that a^b * a^c = a^(b+c)?
......!!!!!!!!...................................
RESPONSE --> a^b*a^c say b=2 and c=3, this would turn into a*a * a*a*a which is equal to a^5, the exponents may be added together if the base is the same.
.................................................
......!!!!!!!!...................................
22:52:23 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).
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:53:48 `q004. What does (2^3)^5 mean?
......!!!!!!!!...................................
RESPONSE --> (2^3)^5 = (2*2*2)(2*2*2)(2*2*2)(2*2*2)(2*2*2) = 2^15 = 32768
.................................................
......!!!!!!!!...................................
22:54:02 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:55:19 `q005. Why do we say that (a^b)^c = a^(b*c)?
......!!!!!!!!...................................
RESPONSE --> As from the previous question, (2^3)^5 turns into (2*2*2)(2*2*2)(2*2*2)(2*2*2)(2*2*2) or 2^15, so whenever there is (a^b)^c it can be reduced to a^(b*c)
.................................................
......!!!!!!!!...................................
22:55:37 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).
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:56:43 `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?
......!!!!!!!!...................................
RESPONSE --> 2^5 * 2^-2 = 2^(5-2) = 2^3
.................................................
......!!!!!!!!...................................
22:56:45 To according to the law, 2^5 * 2^-2 = 2^(5 + -2) = 2^(5-2) = 2^3.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
22:58:52 `q007. Since as we saw in the preceding question 2^5 * 2^-2 = 2^3, what therefore must be the value of 2^-2?
......!!!!!!!!...................................
RESPONSE --> 2^(-2) = .25 to satisfy 2^5* 2^-2 = 2^3 32*.25 = 8
.................................................
......!!!!!!!!...................................
22:59:00 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:00:06 `q008. Why must we say that 2^-n = 1 / 2^n, where n stands for any counting number?
......!!!!!!!!...................................
RESPONSE --> That is the common rule for a negative power, I cannot say why any further then that.
.................................................
......!!!!!!!!...................................
23:00:58 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.
......!!!!!!!!...................................
RESPONSE --> I have never seen it put that way.
.................................................
......!!!!!!!!...................................
23:02:40 `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?
......!!!!!!!!...................................
RESPONSE --> 2^3 * 2^-3 = 2^(3-3) = 2^0 = 1 2^3 * (1/2^3) = 8*1/8 = 1
.................................................
......!!!!!!!!...................................
23:02:44 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:02:51 `q010. Continuing the last question, what therefore should be the value of 2^0?
......!!!!!!!!...................................
RESPONSE --> 1
.................................................
......!!!!!!!!...................................
23:02:54 Since 2^3 * 2^-3 = 2^0 and also 2^3 * 2^-3 = 1 we see that 2^0 must be 1.
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:04:31 `q011. How do we solve the equation x^3 = 12?
......!!!!!!!!...................................
RESPONSE --> to solve x^3 = 12, you would multiply both sides by the reciprocal of the power three, so that would be 1/3, this cancels out x raised to the third power to give x=12^1/3
.................................................
......!!!!!!!!...................................
23:04:38 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 .
......!!!!!!!!...................................
RESPONSE -->
.................................................
......!!!!!!!!...................................
23:06:04 `q012. How do we solve the equation x^(5/2) = 44?
......!!!!!!!!...................................
RESPONSE --> You would multiply both sides of x^(5/2) = 44 by the reciprocal 2/5, so x=44^2/5 = approx. 4.54
.................................................
......!!!!!!!!...................................
23:06:08 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.
......!!!!!!!!...................................
RESPONSE -->
.................................................
"