Question

Solve this equation, and show work. worth 40 points.

Answer 1

`b=-3`

Explanation:

We have:

`25^b\cdot 125^(b+2)=125^b`

Notice that 125 is the same as 5³. So:

`25^b\cdot ((5)^3)^(b+2)=((5)^3)^b`

Also, 25 is the same as 5². So:

`((5)^2)^b\cdot ((5)^3)^(b+2)=((5)^3)^b`

Power of a power property:

`5^(2b)\cdot 5^(3(b+2))=5^(3b)`

Power of a product property:

`5^(2b+3(b+2))=5^(3b)`

Since they have the same base, the exponents are equal. So:

`2b+3(b+2)=3b`

Solve for b. Distribute on the left:

`2b+3b+6=3b`

Combine like terms on the left:

`5b+6=3b`

Subtract 5b from both sides:

`6=-2b`

Divide both sides by -2. So, the value of b is:

`b=-3`

And we're done!