To solve the given problem and simplify the expression:
[tex]\[
\frac{3x^2}{x^2 - 16} \div \frac{x^5}{(x - 4)^2}
\][/tex]
we can work through the following steps:
### Step 1: Understand Division of Fractions
To divide fractions, recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, the given expression can be rewritten as:
[tex]\[
\frac{3x^2}{x^2 - 16} \div \frac{x^5}{(x - 4)^2} = \frac{3x^2}{x^2 - 16} \times \frac{(x - 4)^2}{x^5}
\][/tex]
### Step 2: Expand and Simplify
Now we need to simplify the expression before multiplying:
- The denominator [tex]\( x^2 - 16 \)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - 16 = (x - 4)(x + 4)
\][/tex]
So the expression becomes:
[tex]\[
\frac{3x^2}{(x - 4)(x + 4)} \times \frac{(x - 4)^2}{x^5}
\][/tex]
### Step 3: Multiply the Fractions
Next, multiply the numerators together and the denominators together:
[tex]\[
\frac{3x^2 \cdot (x - 4)^2}{(x - 4)(x + 4) \cdot x^5}
\][/tex]
### Step 4: Simplify the Resulting Fraction
Combine and simplify the terms:
[tex]\[
\frac{3x^2 (x - 4)^2}{x^5 (x - 4)(x + 4)}
\][/tex]
Cancel the common factors in the numerator and the denominator. The [tex]\( x^2 \)[/tex] and [tex]\( x^5 \)[/tex] terms can be reduced, and one [tex]\( (x - 4) \)[/tex] term from the numerator and denominator can be canceled:
[tex]\[
\frac{3 (x - 4)}{x^3 (x + 4)}
\][/tex]
### Step 5: Final Simplified Expression
Combining and simplifying all terms, we obtain the final simplified form:
[tex]\[
\frac{3(x - 4)}{x^3(x + 4)}
\][/tex]
Therefore, the simplified quotient is:
[tex]\[
\frac{3(x - 4)}{x^3(x + 4)}
\][/tex]
