Answer:
x = 7
Explanation:
To solve for x in this equation, we're going to need to get the two exponents (3x - 5 and x + 1) equal to each other, but we can't do that unless our bases are the same.
For example, in [tex]A^{x} = B^{x + 1}[/tex], you cannot solve x = x + 1. In [tex]A^{x} = A^{x + 1}[/tex], you can solve x = x + 1.
Just by looking at the bases, 5 and 25, you can tell that it will be simple to make them match. 25 is just 5². The tricky part is going to be figuring out where to put the 2 into x + 1.
Let's look at another example. If you have [tex]2^{2 * 2}[/tex], then you can simplify it to [tex]2^{4}[/tex], which is 16. Or, you could do them one at a time, so [tex](2^{2}) ^{2}[/tex]. This way you'd have 4², which you'd be able to recognize as 16. Based on this example, we know that to make our bases the same, we need to change [tex]25^{x + 1}[/tex] to [tex]5^{2(x + 1)}[/tex].
[tex]5^{3x-5} = 25^{x+1}[/tex] Change the right side to [tex]5^{2(x + 1)}[/tex]
[tex]5^{3x-5} = 5^{2(x+1)}[/tex] Simplify that exponent using distribution
[tex]5^{3x-5} = 5^{2x+2}[/tex]
Now that the bases match, you can get rid of them and just set the exponents equal to each other and solve for x.
3x - 5 = 2x + 2 Add 5 to both sides
3x = 2x + 7 Subtract 2x from both sides
x = 7
Now, check you work!
[tex]5^{3x-5} = 25^{x+1}[/tex] Plug in 7 for x
[tex]5^{3(7)-5} = 25^{(7)+1}[/tex] Simplify
[tex]5^{21-5} = 25^{8}[/tex] Simplify one more time
[tex]5^{16} = 25^{8}[/tex] Plug these into a calculator if you have one
152587890625 = 152587890625 So you know that x = 7 is correct.
