Solve: Multiplying Rational Expressions with (16x⁴)/(5y) and (10y²)/(3x⁴y)

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 the problem and perform the multiplication between the fractions:

$\frac{16x^4}{5y}\cdot\frac{10y^2}{3x^4y}=\frac{16\cdot10\cdot x^4y^2}{5\cdot3\cdot x^4y\cdot y}=\frac{160x^4y^2}{15x^4y^2}$

Where 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 using the distributive property of multiplication and the law of exponents for multiplying terms with identical bases:

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

Which we applied in the last stage to the denominator of the resulting fraction.

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:

$\frac{160x^4y^2}{15x^4y^2}=\frac{160}{15}\cdot\frac{x^4}{x^4}\cdot\frac{y^2}{y^2}$

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 this law to the last expression we got:

$\frac{160}{15}\cdot\frac{x^4}{x^4}\cdot\frac{y^2}{y^2}=\frac{32}{3}\cdot x^{4-4}\cdot y^{2-2}=\frac{32}{3}x^0y^0$

Where in the first stage, in addition to applying the above law of exponents, we also simplified the numerical fraction after identifying that both its numerator and denominator are multiples of 5, and then simplified the resulting expression,

In the next stage we'll recall that raising any number to the power of 0 gives the result 1, meaning mathematically that:

$X^0=1$

Let's return to the expression we got and continue simplifying using this fact:

$\frac{32}{3}x^0y^0=\frac{32}{3}\cdot1\cdot1=\frac{32}{3}$

We can now convert the improper fraction we got to a mixed number to get:

$\frac{32}{3}=10\frac{2}{3}$

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

$\frac{16x^4}{5y}\cdot\frac{10y^2}{3x^4y}=\frac{160x^4y^2}{15x^4y^2}=10\frac{2}{3}$

Therefore the correct answer is answer C.

Important Note:

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

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

$\frac{16x^4}{5y}\cdot\frac{10y^2}{3x^4y}=10\frac{2}{3}$

(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 simplified 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 put a single fraction line like we did at the beginning and can apply the distributive property of multiplication and so on, this is a point worth noting, since not every problem we encounter will meet all the conditions mentioned here in this note.