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To fully simplify the expression:

\[
\frac{2 t u^3 w^7}{18 u^{15} w}
\]

we'll go through each component separately.

1. **Simplify the coefficients**:
   \[
   \frac{2}{18} = \frac{1}{9}
   \]

2. **Simplify the \( u \) terms**:
   - In the numerator, we have \( u^3 \).
   - In the denominator, we have \( u^{15} \).
   - Using the rule \( \frac{a^m}{a^n} = a^{m - n} \), we get:
     \[
     u^{3 - 15} = u^{-12} = \frac{1}{u^{12}}
     \]

3. **Simplify the \( w \) terms**:
   - In the numerator, we have \( w^7 \).
   - In the denominator, we have \( w \) (which is \( w^1 \)).
   - Using the same rule, we get:
     \[
     w^{7 - 1} = w^6
     \]

4. **Combine all simplified components**:
   - After simplification, we have:
     \[
     \frac{t w^6}{9 u^{12}}
     \]

So, the fully simplified expression is:

\[
\frac{t w^6}{9 u^{12}}
\]

Would you like further details, or have any questions?

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Here are some related questions to expand your understanding:

1. How do you handle negative exponents in simplification problems?
2. What is the general rule for simplifying terms with the same base in fractions?
3. How do you apply exponent rules when simplifying expressions with multiple variables?
4. How can you check if an expression is fully simplified?
5. What are the steps to simplify an expression with fractional coefficients?

**Tip:** Remember to simplify coefficients first before handling variables when simplifying expressions in fraction form.

Fully simplify the expression \( \frac{2tu^3 w^7}{18u^{15}w} \) and give the answer as a fraction without any negative indices.

Learn how to fully simplify the expression \( \frac{2tu^3 w^7}{18u^{15}w} \) by reducing coefficients and applying exponent rules. This problem covers algebraic concepts in simplifying expressions with exponents, suitable for students in Grades 9-10.

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