To simplify the given expression:
[tex]\[
\frac{a^{m} \times b^{n} \times c^{ab}}{m^a \times n^b \times (b^a)^c}
\][/tex]
we can follow these steps:
1. Identify the components in both the numerator and the denominator.
Numerator: [tex]\(a^{m} \times b^{n} \times c^{ab}\)[/tex]
Denominator: [tex]\(m^a \times n^b \times (b^a)^c\)[/tex]
2. Rewrite the denominator’s term [tex]\((b^a)^c\)[/tex].
According to the power of a power property, [tex]\((x^m)^n = x^{mn}\)[/tex]. Thus,
[tex]\((b^a)^c = b^{ac}\)[/tex]
Now the expression in the denominator becomes:
[tex]\(m^a \times n^b \times b^{ac}\)[/tex]
3. Combine the like terms in the numerator and the denominator.
At this step, the original expression can be represented as:
[tex]\[
\frac{a^{m} \times b^{n} \times c^{ab}}{m^a \times n^b \times b^{ac}}
\][/tex]
4. Notice that the base 'b' appears in both a multiplication term in the numerator and the denominator. We can combine these terms using the property of exponents: [tex]\(\frac{x^m}{x^n} = x^{m-n}\)[/tex].
[tex]\[
\frac{a^{m} \times b^{n} \times c^{ab}}{m^a \times n^b \times b^{ac}} = \frac{a^{m} \times b^{n-ac} \times c^{ab}}{m^a \times n^b}
\][/tex]
5. Since there are no more like terms to combine and the bases in the numerator and the denominator are distinct from each other, this is our simplest form:
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{a^{m} \times b^{n-ac} \times c^{ab}}{m^a \times n^b}}
\][/tex]
This is the detailed, step-by-step solution for simplifying the given expression.
