Simplify: ((5x+4y)³·7x·3xy)/(45y·x²·(5x+4y)²) - Complex Fraction Solution

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

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

But before we perform the multiplication of fractions, let's note

Important note-

Notice that in both fractions in the multiplication there are expressions that are binomials squared

Of course, we're referring to-$(5x+4y)^3$ and$(5x+4y)^2$,

We'll treat these expressions simply as terms squared - meaning:

We won't open the parentheses and we'll treat both expressions exactly as we treat terms:

$a^3$and$a^2$.

Now let's return to the problem and continue from where we left off:

$$Let's apply the above-mentioned rule for multiplying fractions in the problem and perform the multiplication between the fractions:

$\frac{(5x+4y)^3\cdot7x}{45y\cdot x^2}\cdot\frac{3xy}{(5x+4y)^2}=\frac{7\cdot3xxy(5x+4y)^3}{45\cdot x^2y(5x+4y)^2}=\frac{7x^2y(5x+4y)^3}{15x^2y(5x+4y)^2}$

where in the first stage we performed the multiplication between the fractions using the above rule, and in the second stage we reduced the numerical part in the resulting fraction, then we 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 final stage to the numerator and denominator of the resulting fraction.

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

$\frac{7x^2y(5x+4y)^3}{15x^2y(5x+4y)^2}=\frac{7}{15}\cdot\frac{x^2}{x^2}\cdot\frac{y}{y}\cdot\frac{(5x+4y)^3}{(5x+4y)^2}=\frac{7}{15}\cdot1\cdot1\cdot\frac{(5x+4y)^3}{(5x+4y)^2}=\frac{7}{15}\cdot\frac{(5x+4y)^3}{(5x+4y)^2}$

where additionally in the second stage we applied the fact that dividing any number by itself gives 1.

We can now 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:

$\frac{7}{15}\cdot\frac{(5x+4y)^3}{(5x+4y)^2}=\frac{7}{15}\cdot(5x+4y)^{3-2}=\frac{7}{15}\cdot(5x+4y)^{1}=\frac{7}{15}\cdot(5x+4y)$

where in the first stage we applied the above law of exponents, treating the binomial as noted above, and then simplified the resulting expression remembering that raising a number to the power of 1 gives the number itself,

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

$\frac{(5x+4y)^3\cdot7x}{45y\cdot x^2}\cdot\frac{3xy}{(5x+4y)^2}= \frac{7}{15}\cdot\frac{(5x+4y)^3}{(5x+4y)^2} =\frac{7}{15}\cdot(5x+4y)$

Therefore the correct answer is answer D.

Another 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 reduction of the numerical part to get directly the last line we got:

$\frac{(5x+4y)^3\cdot7x}{45y\cdot x^2}\cdot\frac{3xy}{(5x+4y)^2}=\frac{7}{15}\cdot(5x+4y)$

(meaning we could have skipped the part where we expressed the fraction as a multiplication of fractions and even the initial multiplication of fractions we performed and gone straight to reducing 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 of the fractions in the problem, and also between the fractions themselves, multiplication is performed, meaning, that we can combine into one unified fraction as we did at the beginning and can apply the distributive property of multiplication and express as multiplication of fractions as mentioned above, etc., this is a point worth noting, since not in every problem we encounter all the conditions mentioned here in this note are met.