x^4 + x^5 = 100

		Quandaries and Queries
	Name: Bill Who is asking: Teacher Level: Secondary Question: I'm a high school math teacher, so maybe I shouldn't be owning up to this, but... One of my students has stumped me. He asked how to solve the equation 4^x + 5^x = 100 All I can think of are graphing methods to get an approximate solution. What am I missing? Thanks.
	Hi Bill, One of the exciting things that happens sometimes to us who are teachers is when a student stumps us with a question we can't answer. It shows that you have challenged the student to think beyond what you are doing. Your suggestion of a graphical method to approximate a solution is good. With this problem the best you can do is approximate a solution. You could also try a numeric approximation using a binary search. Start with two approximations for x, one which is too large and the other too small. I used 2 and 3 4² + 5² = 41 4³ + 5³ = 189 For the next estimation for x use the number which is half way between 2 and 3, that is x = ^(2 + 3)/₂ = 2.5 This gives 4^2.5 + 5^2.5 = 87.09 Hence x = 2.5 is still too small but now you know that the true value of x is between 2.5 and 3 so try x = ^(2.5 + 3)/₂ = 2.75 This time you get 4^2.75 + 5^2.75 = 128.85 Again too large but now you know that 2.5 < x < 2.75 and hence try x = ^{(2.5 + 2.75)}/₂ = 2.625 4^2.625 + 5^2.625 = 106.41 so now you know that 2.5 < x < 2.625 What you arer doing here is finding an interval estimate for x, and at each step the interval is one half the length of the previous interval. If your student has a way to write a program you might suggest that he implement this algorithm. Even without a way to program it you can do it on a calculator or spread sheet. If the student can program it have him use the same algorithm to approximate the square root of 2 and ask him how many steps it takes to get the answer correct to 5 significant digits. I hope this helps, Harley
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