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/