Question

What is the value of X?

Answer 1

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 $A^{x} = B^{x + 1}$, you cannot solve x = x + 1. In $A^{x} = A^{x + 1}$, 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 $2^{2 * 2}$, then you can simplify it to $2^{4}$, which is 16. Or, you could do them one at a time, so $(2^{2}) ^{2}$. 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 $25^{x + 1}$ to $5^{2(x + 1)}$.

$5^{3x-5} = 25^{x+1}$   Change the right side to $5^{2(x + 1)}$

$5^{3x-5} = 5^{2(x+1)}$   Simplify that exponent using distribution

$5^{3x-5} = 5^{2x+2}$  

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!

$5^{3x-5} = 25^{x+1}$   Plug in 7 for x

$5^{3(7)-5} = 25^{(7)+1}$   Simplify

$5^{21-5} = 25^{8}$   Simplify one more time

$5^{16} = 25^{8}$   Plug these into a calculator if you have one

152587890625 = 152587890625   So you know that x = 7 is correct.

Answer 2

Answer:

x = 7

Explanation:

Because 25 is a perfect square of 5, we can turn

$5^{3x - 5}$ = $25^{x + 1}$

into

$5^{3x - 5}$ = $5^{2x + 2}$

Since the bases are now both equal, we can completely ignore them, as we are only trying to find x. This leaves us with:

3x - 5 = 2x + 2

All we have to do now is solve for x:

x - 5 = 2          Subtract 2x from both sides.

x = 7                Add 5 to both sides.

Hope this helps! :)