Solve: (38x⁵y⁴/9x) × (5xy/3y²) Fraction Multiplication

Let's start with multiplying the two fractions in the problem using the rule for fraction multiplication, which states that we multiply numerator by numerator and denominator by denominator while keeping the fraction line:

$\frac{a}{b}\cdot\frac{c}{d}=\frac{a\cdot c}{b\cdot d}$Let's apply this rule to our problem and perform the multiplication between the fractions:

$\frac{38\cdot x^5y^4}{9x}\cdot\frac{5xy}{3y^2}=\frac{38\cdot5\cdot x^5xy^4y}{9\cdot3\cdot xy^2}=\frac{190\cdot x^6y^5}{27\cdot xy^2}$

In the first stage, we performed the multiplication between the fractions using the above rule, and then simplified the expressions in the numerator and denominator of the resulting fraction by using the distributive property of multiplication and the law of exponents for multiplying terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

We applied this in the final stage to the fraction's numerator.

Now we'll use the same rule for fraction multiplication again, but in the opposite direction, in order to express the resulting fraction as a multiplication of fractions where each fraction contains only numbers or terms with identical bases:

$\frac{190\cdot x^6y^5}{27\cdot xy^2}=\frac{190}{27}\cdot\frac{x^6}{x}\cdot\frac{y^5}{y^2}$

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

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

Let's apply the above law to the last expression we got:

$\frac{190}{27}\cdot\frac{x^6}{x}\cdot\frac{y^5}{y^2}=\frac{190}{27}x^{6-1}y^{5-2}=\frac{190}{27}\cdot x^5y^3=7\frac{1}{27}\cdot x^5y^3$

In the first stage we applied the above law of exponents, then simplified the resulting expression, additionally we removed the multiplication sign and switched to the conventional multiplication notation by placing the terms next to each other, and in the final stage we converted the improper fraction we got at the beginning of the last expression to a mixed number.

Let's summarize the solution to the problem, we got that:

$\frac{38\cdot x^5y^4}{9x}\cdot\frac{5xy}{3y^2}= \frac{190\cdot x^6y^5}{27\cdot xy^2} =7\frac{1}{27}\cdot x^5y^3$

Therefore the correct answer is answer B.

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 numerical part reduction to get directly the last line we received:

$\frac{38\cdot x^5y^4}{9x}\cdot\frac{5xy}{3y^2}=7\frac{1}{27}\cdot x^5y^3$

(Meaning we could have skipped the part where we expressed the fraction as a multiplication of fractions and even the initial fraction multiplication we performed and immediately perform the reduction between the fractions)

However, it should be emphasized that this quick solution method is conditional on the fact that between all terms in the numerator and denominator of each fraction in the problem, and also between the fractions themselves, multiplication is performed, meaning that we can enter a single fraction line as we did at the beginning and can apply the distributive property and express as fraction multiplication etc., this is a point worth noting, since not every problem we encounter will meet all the conditions mentioned here in this note.