What is the product of the expressions? Assume [tex] y \neq 0 [/tex].

[tex]\[ \left( \frac{3x + 5}{y^5} \right) \left( \frac{4y^3}{5} \right) \][/tex]

A. [tex]\[ \frac{12x}{5y^2} \][/tex]

B. [tex]\[ \frac{12x + 20}{5y^2} \][/tex]

C. [tex]\[ \frac{12x}{5y^3} \][/tex]

D. [tex]\[ \frac{12x + 20}{5y^3} \][/tex]



Answer :

To find the product of the given expressions [tex]\(\left(\frac{3x + 5}{y^5}\right) \left(\frac{4y^3}{5}\right)\)[/tex], we will use the properties of fractions and multiplication. Here's a step-by-step solution:

1. Multiply the numerators:
The numerators are [tex]\(3x + 5\)[/tex] and [tex]\(4y^3\)[/tex].
[tex]\[ (3x + 5) \cdot 4y^3 = 4y^3(3x + 5) = 12xy^3 + 20y^3 \][/tex]

2. Multiply the denominators:
The denominators are [tex]\(y^5\)[/tex] and [tex]\(5\)[/tex].
[tex]\[ y^5 \cdot 5 = 5y^5 \][/tex]

3. Combine the results:
We can now write the product of the expressions as:
[tex]\[ \frac{12xy^3 + 20y^3}{5y^5} \][/tex]

4. Simplify the fraction:
We will divide both terms in the numerator by the denominator:
[tex]\[ \frac{12xy^3}{5y^5} + \frac{20y^3}{5y^5} \][/tex]

Simplify each term:
[tex]\[ \frac{12xy^3}{5y^5} = \frac{12x}{5y^2} \][/tex]
[tex]\[ \frac{20y^3}{5y^5} = \frac{20}{5y^2} = \frac{4}{y^2} \][/tex]

5. Combine the simplified terms:
[tex]\[ \frac{12x}{5y^2} + \frac{4}{y^2} \][/tex]

6. Factor out the common term [tex]\(\frac{1}{y^2}\)[/tex]:
[tex]\[ \frac{12x + 4}{5y^2} \][/tex]

Thus, the simplified form of the product of the expressions is:
[tex]\[ \frac{12x + 20}{5y^2} \][/tex]

So the correct answer is:
[tex]\[ \boxed{\frac{12x + 20}{5y^2}} \][/tex]