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Let's simplify the given expression step-by-step:

The expression is:

\[
\frac{16t^2 y^6}{t + 7y} \div \frac{4t^6 y}{t^2 - 49y^2}
\]

### Step 1: Convert division to multiplication
We know that dividing by a fraction is the same as multiplying by its reciprocal. So, the expression becomes:

\[
\frac{16t^2 y^6}{t + 7y} \times \frac{t^2 - 49y^2}{4t^6 y}
\]

### Step 2: Factorize the denominator \( t^2 - 49y^2 \)
Notice that \( t^2 - 49y^2 \) is a difference of squares, so it can be factored as:

\[
t^2 - 49y^2 = (t - 7y)(t + 7y)
\]

Substitute this into the expression:

\[
\frac{16t^2 y^6}{t + 7y} \times \frac{(t - 7y)(t + 7y)}{4t^6 y}
\]

### Step 3: Cancel common terms
The \( t + 7y \) term appears in both the numerator and the denominator, so we can cancel it:

\[
\frac{16t^2 y^6}{1} \times \frac{(t - 7y)}{4t^6 y}
\]

Now simplify the remaining terms.

### Step 4: Simplify coefficients and powers of \( t \) and \( y \)
- Coefficients: \( \frac{16}{4} = 4 \)
- Powers of \( t \): \( \frac{t^2}{t^6} = \frac{1}{t^4} \)
- Powers of \( y \): \( \frac{y^6}{y} = y^5 \)

This simplifies the expression to:

\[
4 \cdot \frac{(t - 7y)}{t^4} \cdot y^5
\]

### Step 5: Distribute and express the result
Distribute \( 4y^5 \) over \( t - 7y \):

\[
\frac{4y^5(t - 7y)}{t^4}
\]

This can be written as:

\[
\frac{4y^5 t}{t^4} - \frac{4y^6}{t^4}
\]

### Step 6: Final simplified form
Simplify each term:

\[
= \frac{4y^5}{t^3} - \frac{4y^6}{t^4}
\]

This is the simplified expression.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How can we simplify expressions involving differences of squares?
2. What are some general tips for factoring complex polynomials?
3. How do you divide fractions and convert them to multiplication?
4. Can you identify other types of common factoring patterns?
5. What are the rules for simplifying powers of variables during division?

**Tip**: Always check for common factors across numerators and denominators to simplify expressions quickly!

Simplify the expression: (16t^2 y^6) / (t + 7y) ÷ (4t^6 y) / (t^2 - 49y^2)

This problem simplifies a rational expression involving exponents and factoring. It involves multiplying and dividing rational expressions, factoring differences of squares, and simplifying powers of variables. Suitable for students in Grades 9-11.

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