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if an initial investment P bears an int bears interest at a rate R so this is initial investment P bears interest at a rate R and is compounded annually its future value a after T years can be determined with the equation above which of the following equations shows the interest rate in terms of the future value initial investment and number of years invested so they really just want us to solve for R so let's see if we can do that so we have the future value let me give myself some space future value is equal I'm just rewriting it is equal to the initial investment times 1 plus our interest rate and then that quantity to the T power so let's see what we can do so the first thing I could do is I can get the P on to the left hand side by dividing both sides of this equation by our initial investment so if I do that I get and actually let me let me swap the sides to then I get 1 plus R to the T power is equal to our future value divided by our initial investment now how do I get rid of this to the T power here well I could raise both sides to the 1 over t power so I could raise that to the 1 over t power 1 over T if I do it on the left side I have to do it to the right side as well so on the left hand side if I raise something to the T and then I raise it to the 1 over T remember if you raise something to an exponent and then raise it to another exponent you're raising it to the product of these two exponents so this is equivalent to raising 1 this is equivalent to just raising 1 plus R to the first power or this is just going to simplify to 1 plus R on the left hand side and on the right hand side we're gonna have our future value divided by our initial investment to the 1 over t power and now pretty straightforward you want to solve for R subtract 1 from both sides we get R is equal to our future value divided by our initial investment to the 1 over tea power minus one and that is this choice right over there