Solve the Algebraic Fraction: Simplify 78xy^5 Over 3x^5 Times 4yx Over 5y^4

To solve the problem, we'll follow these steps:

• Step 1: Multiply the fractions $\frac{78xy^5}{3x^5}$ and $\frac{4yx}{5y^4}$.

• Step 2: Simplify the coefficients and apply exponent rules to the variables.

• Step 3: Identify the correct multiple-choice option matching the simplified expression.

Now, let's work through each step in detail:

Step 1: Multiply the Fractions

We multiply the numerators together and the denominators together:

$\frac{78xy^5}{3x^5} \cdot \frac{4yx}{5y^4} = \frac{78xy^5 \cdot 4yx}{3x^5 \cdot 5y^4}$

Simplifying, we have:

$= \frac{78 \cdot 4 \cdot x \cdot x \cdot y^5 \cdot y}{3 \cdot 5 \cdot x^5 \cdot y^4}$

Step 2: Simplify the Expression

Simplify the coefficients:

$78 \cdot 4 = 312$ and $3 \cdot 5 = 15$

Combine the coefficients:

$\frac{312}{15}$

Now simplify the variables using exponent rules:

Combine powers of the same base:

$x \cdot x = x^2$

The numerator becomes:

$312 \cdot x^2 \cdot y^6$

The denominator according to $\frac{x^2}{x^n} = x^{2-n}$, given,

$15 \cdot x^5 \cdot y^4$

Simplifying the exponents:

$x^{2-5} = x^{-3}$ and $y^{6-4} = y^2$

Thus,

$\frac{312 \cdot x^{-3} \cdot y^2}{15}$

Conclusion:

After simplifying the expression, the result is:

$\frac{312\cdot x^{-3}\cdot y^2}{15}$

Matching this with the multiple-choice options, the correct choice is option 3.

Therefore, the solution to the problem is $\frac{312\cdot x^{-3}\cdot y^2}{15}$.