Question

Solve the equation (linear equation)


`8^(2x+7) = ((1)/(32))^(3x)`

Answer 1

Answer:
`x=-1`

Explanation:

By the negative exponent rule, you have that:

`((1)/(a))^n=a^(-n)`

By the exponents properties, you know that:

`(m^n)^l=m^((nl))`

Therefore, you can rewrite the left side of the equation has following:

`((1)/(8))^(-(2x+7))=((1)/(32))^(3x)`

Descompose 32 and 8 into its prime factors:

`32=2*2*2*2*2=2^5\\8=2*2*2=2^3`

Rewrite:

`((1)/(2^3))^(-(2x+7))=((1)/(2^5))^(3x)`

Then:

`((1)/(2))^(-3(2x+7))=((1)/(2))^(5(3x))`

As the base are equal, then:

`-3(2x+7)=5(3x)`

Solve for x:

`-6x-21=15x\\-21=15x+6x\\-21=21x\\x=-1`