Need help finding x.

Question

6^x+8^x=10^x

Please show me how it's done.

Answer 1

I can give you a solution that involves some guessing.

6^x = [2(3)]^x = 2^x3^x

8^x = [2(4)]^x = 2^x4^x

10^x = [2(5)]^x = 2^x5^x

So now we have

2^x3^x + 2^x4^x = 2^x5^x

Divide both sides by 2^x.

3^x + 4^x = 5^x

Now compare this to the Pythagorean trio of

3² + 4² = 5²

It looks like

x = 2

Test this answer.

6² + 8² = 10² (good)