Question

Solve the exponential equation by rewriting the base. explain the steps towards your answer.


`((1)/(9) )^(2n) =27^(n)`

Answer 1

n=0

Explanation:

Given in the question an equation

`(1)/(9)^(2n)=27^(n)`

we can write

`(1)/(9)`

as

`((1)/(3))^(2)`

further as

`3^(-2)`

so

`3^(-2(2n))=3^(3n)`

`3^(-4n)=3^(3n)`

Since now the base is same

and we know that

`x^(n)=x^(m)\\n=m`

-4n = 3n

n cancel out which means n =0

Answer 2

Answer:
`n=0`

Explanation:

You need to remember that:

`a^n=a^m\\n=m`

By the Power of a power property:

`(a^m)^n=a^(mn)`

By the Negative exponent rule:

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

Therefore, having the expression
`((1)/(9) )^(2n)=27^(n)`:

Descompose 27 and 9 into their prime factors:

`27=3*3*3=3^3`

`9=3*3=3^2`

Rewrite the expression and simplify. Then:

`(3^(-2))^(2n)=(3^3)^n\\3^(-4n)=3^(3n)\\-4n=3n\\-4n-3n=0\\n=0`