Simplify the Algebraic Expression: Tackling 15x⁴y³/8x²y⁵ x 24yx⁷/3xy²

To solve this problem, we'll proceed with the following steps:

• Step 1: Simplify each fraction separately.

Consider the first fraction:

$\frac{15x^4y^3}{8x^2y^5}$

Apply the quotient rule of exponents: $\frac{x^m}{x^n} = x^{m-n}$ and $\frac{y^m}{y^n} = y^{m-n}$.

This gives us: $\frac{15}{8} \cdot x^{4-2} \cdot y^{3-5} = \frac{15}{8} \cdot x^2 \cdot y^{-2}$.

• Step 2: Simplify the second fraction.

Consider the second fraction:

$\frac{24yx^7}{3xy^2}$

Apply the quotient rule: $\frac{24}{3} \cdot \frac{y}{y^2} \cdot \frac{x^7}{x^1} = 8 \cdot y^{1-2} \cdot x^{7-1} = 8 \cdot y^{-1} \cdot x^6$.

• Step 3: Multiply the simplified fractions together.

Now, multiply the results:

$\left(\frac{15}{8} \cdot x^2 \cdot y^{-2}\right) \cdot \left(8 \cdot y^{-1} \cdot x^6\right)$

Simplify by multiplying coefficients and applying exponent rules: $\frac{15 \times 8}{8} \cdot x^{2+6} \cdot y^{-2-1}$.

Which simplifes to: $15 \cdot x^8 \cdot y^{-3}$.

Therefore, the expression simplifies to $15x^8y^{-3}$.

Finally, matching this result with the provided choices, we find that the correct answer is choice (3):

$15x^8y^{-3}$