Finding Roots on a Calculator
Date: 04/22/2001 at 04:40:52 From: Matt Moore Subject: Cube roots I'm trying to solve an "Annual Holding Period Return" problem. I have already been provided with the answer, but I do not have a cube root function on my calculator. Here's the equation: Annual HPR = (1.667)^(1/5) = 1.1076 If I had the cube root function, I would have just entered 1.667 and hit the cube root key once and then the squared key once. Is that the right way to solve this problem, or is there a quicker way to arrive at the answer? Can I solve this problem with a calculator that doesn't have a cube root function? I have one more question. How do I arrive at the following answer? 1.133^(1/.75) = 1.1816 Thank you, Matt
Date: 04/24/2001 at 02:35:11 From: Doctor Douglas Subject: Re: Cube roots Hi Matt, and thanks for writing to Ask Dr. Math. Since you are raising to the (1/5) power, you need the fifth root, not the cube root. Note that if you hit "cube-root" and "squared" in succession, you obtain the quantity raised to the 2/3 power. The 2 comes from the "squared" and the 1/3 comes from the cube root. If you hit "cube-root" and "square-root" in succession, you obtain the quantity raised to the 1/6 (= 1/3 * 1/2) power. You actually need the "fifth-root" button - but that's not a very common button to find on a calculator! But there is a way to calculate the above quantity. On a calculator with the y^x key, you can key in something like the following: 1.66666667 y^x ( 1 / 5 ) = \___\__\___\____ "raise y to the power x" \__\___\___ open parenthesis \___\__ one divided by five \_ close parenthesis. Result: 1/5 = 0.2 Final result: 1.66666667^0.2 = 1.1076 If you have the keys log and 10^x, or ln and e^x, you can also obtain this result by taking the log of both sides: log 1.66666667^(1/5) = (1/5) * log(1.6666667) = 0.04436975 Once you obtain this number, you raise 10 to it: 10^0.04436975 = 1.1076 The procedure for ln and e^x is similar and gives the same result. To get the answer 1.133^(1/.75) = 1.1816, again you must use either the y^x key, or one of the following combinations: (10^x and log), (e^x and ln). In the case that the root is a simple number (such as the fifth root), and if we don't have any of the functions listed above, then an alternative is to use trial and error: for example, in your first example above, if we take the number 1.1 and raise it to the fifth power: 1.1 * 1.1 * 1.1 * 1.1 * 1.1 = 1.61051 while 1.2 * 1.2 * 1.2 * 1.2 * 1.2 = 2.48832 we see that 1.66666667 lies between these two values, so that the fifth root lies somewhere between 1.1 and 1.2 (and probably somewhat closer to 1.1). We can take various values between 1.1 and 1.2, and by calculating their fifth powers, we can narrow in on the desired solution. I hope this helps answer your question. Please write back if you need more explanation. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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