Solve: (8x⁷y³/20) × (4/2x⁵y²) - Fraction Multiplication with Variables

Let’s begin by multiplying the two fractions using the rule of fraction multiplication: multiply the numerators together and the denominators together, while keeping the fraction bar in place.

$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$

Let's apply this rule to our problem and perform the multiplication between the fractions:

$\frac{8x^7y^3}{20}\cdot\frac{4}{2x^5y^2}=\frac{8\cdot4\cdot x^7y^3}{20\cdot2\cdot x^5y^2}=\frac{32x^7y^3}{40x^5y^2}$

In the first step, we multiplied the fractions using the rule above and then simplified the numerator and denominator of the resulting fraction.

Next, we apply the same rule in reverse, rewriting the fraction as a product of separate fractions, each containing only numbers or terms with the same base:

$\frac{32x^7y^3}{40x^5y^2}=\frac{32}{40}\cdot\frac{x^7}{x^5}\cdot\frac{y^3}{y^2}$

We did this so we could continue and simplify the expression using the law of exponents for division between terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Proceed to apply this law to the last expression that we obtained:

$\frac{32}{40}\cdot\frac{x^7}{x^5}\cdot\frac{y^3}{y^2}=\frac{4}{5}\cdot x^{7-5}\cdot y^{3-2}=\frac{4}{5}\cdot x^2\cdot y^1=\frac{4}{5}x^2y$

In this step, we not only applied the law of exponents but also simplified the numerical fraction by noticing that both the numerator and denominator are divisible by 8. After simplifying, we rewrote the expression in standard form, removing the multiplication dots and placing the terms side by side.

Finally, we can rewrite the result as a single fraction, recalling that multiplying a number by a fraction is equivalent to multiplying it by the numerator of that fraction:

$\frac{4}{5}x^2y=\frac{4x^2y}{5}$

Let's summarize the solution to the problem, we obtain the following:

$\frac{8x^7y^3}{20}\cdot\frac{4}{2x^5y^2} =\frac{32x^7y^3}{40x^5y^2} =\frac{4x^2y}{5}$

Therefore the correct answer is answer D.

Important note:

In solving the problem above, we detailed the steps to the solution, and used fraction multiplication in both directions and multiple times along with the mentioned law of exponents,

We could have shortened the process, applied the distributive property of multiplication, and performed directly both the application of the mentioned law of exponents and the simplification of the numerical part to get directly the last line we received:

$\frac{8x^7y^3}{20}\cdot\frac{4}{2x^5y^2} =\frac{4x^2y}{5}$

(This means we could have skipped rewriting the fraction as a product of smaller fractions, and even the initial step of multiplying the fractions together. Instead, we could have simplified directly across numerators and denominators.)

However, it’s important to emphasize that this shortcut only works because every term in both numerators and denominators, as well as between the two fractions themselves, is connected by multiplication. This allows us to combine everything under a single fraction bar, simplify, and apply the laws of exponents directly. Not every problem will meet these conditions, so this method should be used with care.