To simplify the expression \(\frac{10x^4 \cdot y}{27x^3 \cdot y^{27}}\), follow these steps:

1. **Simplify the coefficients**:  
   The coefficient is \(\frac{10}{27}\), which cannot be simplified further.

2. **Simplify the \(x\) terms**:  
   You have \(x^4\) in the numerator and \(x^3\) in the denominator.  
   Using the rule \(\frac{x^a}{x^b} = x^{a-b}\), you get:  
   \[
   \frac{x^4}{x^3} = x^{4-3} = x^1 = x
   \]

3. **Simplify the \(y\) terms**:  
   You have \(y\) in the numerator and \(y^{27}\) in the denominator.  
   Again, using the same rule:  
   \[
   \frac{y}{y^{27}} = y^{1-27} = y^{-26}
   \]

4. **Combine everything**:  
   The simplified expression is:
   \[
   \frac{10}{27} \cdot x \cdot y^{-26} = \frac{10x}{27y^{26}}
   \]

So, the simplified expression is \(\frac{10x}{27y^{26}}\).

Would you like more details or have any questions about this solution?

Here are 5 related questions you might consider:

1. How would you simplify \(\frac{15x^5 \cdot y^3}{45x^2 \cdot y^7}\)?
2. What happens if the exponents of \(x\) were both negative in the original expression?
3. How do you simplify expressions involving roots, like \(\frac{\sqrt{x^4}}{\sqrt{x^2}}\)?
4. What is the general rule for simplifying \(\frac{a^m \cdot b^n}{c^p \cdot d^q}\)?
5. How would you handle simplifying the expression if the \(y\) terms had fractional exponents?

**Tip:** Always simplify the coefficients first before moving on to the variables, as it often makes the process clearer and easier.

Learn how to simplify the algebraic expression \frac{10x^4 \cdot y}{27x^3 \cdot y^{27}} step by step using exponent rules. This problem involves key concepts in algebra, such as simplifying terms with exponents and applying fraction rules. Suitable for students in Grades 8-10.

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