To simplify the given expression [tex]\(\frac{25 x^{12} y^{14}}{37 x^5 y^6}\)[/tex], follow these steps:
1. Separate the Coefficients:
The given expression contains constants and variables raised to some powers. The constant term [tex]\(25\)[/tex] in the numerator and [tex]\(37\)[/tex] in the denominator can be separated out.
[tex]\[
\frac{25 x^{12} y^{14}}{37 x^5 y^6} = \frac{25}{37} \cdot \frac{x^{12}}{x^5} \cdot \frac{y^{14}}{y^6}
\][/tex]
2. Apply the Quotient Rule for the Variables:
The quotient rule for exponents states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], where [tex]\(a\)[/tex] is a base, and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are the exponents.
- For [tex]\(x\)[/tex]:
[tex]\[
\frac{x^{12}}{x^5} = x^{12-5} = x^7
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
\frac{y^{14}}{y^6} = y^{14-6} = y^8
\][/tex]
3. Combine the Results:
Now, substitute back the simplified forms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] along with the coefficients,
[tex]\[
\frac{25}{37} \cdot x^7 \cdot y^8
\][/tex]
4. Write the Simplified Expression:
Combining all parts, you get the simplified expression:
[tex]\[
\frac{25 x^7 y^8}{37}
\][/tex]
So, the completely simplified form of the given expression [tex]\(\frac{25 x^{12} y^{14}}{37 x^5 y^6}\)[/tex] is
[tex]\[ \boxed{\frac{25 x^7 y^8}{37}} \][/tex]
