Algebra 2 Properties of exponents

Question

Use the properties of exponents to reveal the approximate equivalent quarterly rate for the following exponential expression (1.3)^t, which has an annual interest rate of 30%.

(a) 1.068%

(b) 6.8%

(c) 7.5%

(d) 1.74%

Answer 1

(1.3)^t represents an (annual) interest rate of 30%. You want to find the quarterly rate that would be equivalent. A quarterly rate would be applied four times in one year. The equation in this case would be 1.3 (a thirty percent increase = (1 +x/100)⁴ (an x percent increase applied 4 times)

the process for solving this can be a little daunting, but some hints, don't bother with the binomial expansion. no one wants to deal with a quartic polynomial. instead, consider taking the fourth root of both sides. or (this is where the properties of exponents comes in) raise both sides to the power of 1/4

1.3 = (1 +x/100)⁴

(1.3)^1/4=((1 +x/100)⁴)^1/4

(1.3)^1/4=1+x/100

This can be solved for x:

(1.3)^1/4-1=x/100

100((1.3)^1/4-1)=x