Question

How do you find the value of c and x that makes the equation `(5^(4x-3))(5^(2x+1))=5^16` true?

Answer 1

x=3

Explanation:

`(5^(4x-3))(5^(2x+1))=5^16`
Notice that the two factors on the left both are powers with the same base 5. The rule for multiplying powers with the same base is "Keep the base, add the exponents." Applying this rule:
`5^((4x-3)+(2x+1))=5^16`
Removing the brackets and adding the like terms in the exponent level on the left side yields
`5^(6x-2)=5^16`
Now, since the bases are the same, 5, on both sides of the equation, the exponents must be equal.
`color(white)(ai)6x-2=16`
`color(white)(aaaaa)6x=16+2`
`color(white)(aaaaa)6x=18`
`color(white)(aaaa)(6x)/6=(18)/6`
`color(white)(aaaaaa)x=3`