Let's solve the given problem step-by-step.
We start with the expression inside the parentheses and simplify it before raising it to the power of 3.
Given expression:
[tex]\[
\frac{m^{10} n^4 y^2}{m^{11} y}
\][/tex]
Step 1: Simplify the expression inside the parentheses.
First, separate the powers of each variable:
[tex]\[
= \frac{m^{10}}{m^{11}} \cdot \frac{n^4}{1} \cdot \frac{y^2}{y}
\][/tex]
Step 2: Simplify each fraction:
- For [tex]\( m \)[/tex]:
[tex]\[
\frac{m^{10}}{m^{11}} = \frac{1}{m^{11-10}} = \frac{1}{m}
\][/tex]
- For [tex]\( n \)[/tex]:
[tex]\[
\frac{n^4}{1} = n^4
\][/tex]
- For [tex]\( y \)[/tex]:
[tex]\[
\frac{y^2}{y} = y^{2-1} = y
\][/tex]
Combining these simplified parts:
[tex]\[
\frac{1}{m} \cdot n^4 \cdot y = \frac{n^4 y}{m}
\][/tex]
Step 3: Raise the simplified expression to the power of 3:
[tex]\[
\left(\frac{n^4 y}{m}\right)^3 = \frac{(n^4 y)^3}{m^3}
\][/tex]
Step 4: Apply the power of 3 to each term inside the numerator:
[tex]\[
= \frac{n^{4 \cdot 3} \cdot y^3}{m^3} = \frac{n^{12} y^3}{m^3}
\][/tex]
So, the simplified form of the given expression raised to the power of 3 is:
[tex]\[
\frac{n^{12} y^3}{m^3}
\][/tex]
Now, let's examine the given expected result expression and simplify it:
[tex]\[
\frac{m^{30} n^{12}}{m^3 y^3}
\][/tex]
Step 5: Simplify the expected result expression:
- Combine the powers of [tex]\( m \)[/tex] in the denominator:
[tex]\[
= \frac{m^{30-3} n^{12}}{y^3} = \frac{m^{27} n^{12}}{y^3}
\][/tex]
Thus, the simplified expected result is:
[tex]\[
\frac{m^{27} n^{12}}{y^3}
\][/tex]
Therefore, after simplifying both expressions, we obtain:
- For the original problem simplified and raised to the power of 3: [tex]\(\frac{n^{12} y^3}{m^3}\)[/tex]
- For the given expected result expression simplified: [tex]\(\frac{m^{27} n^{12}}{y^3}\)[/tex]
Hence the numerical result is:
[tex]\[
(\frac{n^{12} y^3}{m^3}, \frac{m^{27} n^{12}}{y^3})
\][/tex]
