Sure! Let's simplify the expression [tex]\(\frac{24 x^3 + 81 y^3}{4 x + 6 y}\)[/tex] step-by-step.
1. Factor the numerator and denominator if possible:
- Notice that both terms in the denominator [tex]\(4x + 6y\)[/tex] can be simplified by factoring out the greatest common divisor (GCD), which is 2. So, we can write:
[tex]\[
4x + 6y = 2(2x + 3y)
\][/tex]
- For the numerator [tex]\(24x^3 + 81y^3\)[/tex], we note that the terms [tex]\(24x^3\)[/tex] and [tex]\(81y^3\)[/tex] can be grouped to form:
[tex]\[
24x^3 + 81y^3 = 3(8x^3) + 3(27y^3)
\][/tex]
2. Simplify the expression by canceling common factors:
- Substituting back, we get the numerator:
[tex]\[
24x^3 + 81y^3 = 3 \cdot (8x^3) + 3 \cdot (27y^3) = 3(8x^3 + 27y^3)
\][/tex]
And the denominator, as previously factored:
[tex]\[
4x + 6y = 2(2x + 3y)
\][/tex]
3. Combine the rewritten numerator and denominator:
- The expression now is:
[tex]\[
\frac{24x^3 + 81y^3}{4x + 6y} = \frac{3(8x^3 + 27y^3)}{2(2x + 3y)}
\][/tex]
4. Express the simplified fraction:
Since there are no more common factors to cancel in [tex]\(\frac{3(8x^3 + 27y^3)}{2(2x + 3y)}\)[/tex], we obtain the final simplified form:
[tex]\[
\frac{24x^3 + 81y^3}{4x + 6y} = \frac{3(8x^3 + 27y^3)}{2(2x + 3y)}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[\boxed{\frac{3(8x^3 + 27y^3)}{2(2x + 3y)}}\][/tex]
