Certainly! Let's simplify the given expression step-by-step and rewrite it in the form [tex]\(6^n\)[/tex]:
Given expression:
[tex]$
\frac{6^{-6}}{6^{-5}}
$[/tex]
To simplify this, we will use the property of exponents that states when you divide like bases, you subtract the exponents. Specifically, for any base [tex]\(a\)[/tex]:
[tex]$
\frac{a^m}{a^n} = a^{m-n}
$[/tex]
In our case, the base [tex]\(a\)[/tex] is [tex]\(6\)[/tex], the exponent in the numerator [tex]\(m\)[/tex] is [tex]\(-6\)[/tex], and the exponent in the denominator [tex]\(n\)[/tex] is [tex]\(-5\)[/tex].
So, applying the property, we have:
[tex]$
\frac{6^{-6}}{6^{-5}} = 6^{-6 - (-5)}
$[/tex]
Next, we simplify the exponent:
[tex]$
-6 - (-5) = -6 + 5 = -1
$[/tex]
Thus, the expression simplifies to:
[tex]$
6^{-1}
$[/tex]
Therefore, we have:
[tex]$
\frac{6^{-6}}{6^{-5}} = 6^{-1}
$[/tex]
So, the simplified expression in the form [tex]\(6^n\)[/tex] is:
[tex]$
6^{-1}
$[/tex]
