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Answer
\[
\left(5-\frac{a}{2}\right)^3 = 125 - \frac{75a}{2} + \frac{15a^2}{4} - \frac{a^3}{8}
\]
Solution
1. Primeiro, reconhecemos que a expressão é um binômio elevado ao cubo:
\[
\left(5-\frac{a}{2}\right)^3
\]
2. Utilizamos a fórmula do cubo de um binômio:
\[
(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
\]
onde, neste caso, \( x = 5 \) e \( y = \frac{a}{2} \).
3. Calculamos cada termo:
- **Primeiro termo:**
\[
x^3 = 5^3 = 125
\]
- **Segundo termo:**
\[
-3x^2y = -3 \cdot 5^2 \cdot \frac{a}{2} = -3 \cdot 25 \cdot \frac{a}{2} = -\frac{75a}{2}
\]
- **Terceiro termo:**
\[
3xy^2 = 3 \cdot 5 \cdot \left(\frac{a}{2}\right)^2 = 15 \cdot \frac{a^2}{4} = \frac{15a^2}{4}
\]
- **Quarto termo:**
\[
-y^3 = -\left(\frac{a}{2}\right)^3 = -\frac{a^3}{8}
\]
4. Juntando todos os termos, temos:
\[
\left(5-\frac{a}{2}\right)^3 = 125 - \frac{75a}{2} + \frac{15a^2}{4} - \frac{a^3}{8}
\]
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Simplify this solution Extra Insights
Did you know that the expression \( \left(5-\frac{a}{2}\right)^{3} \) is a perfect candidate for the Binomial Theorem? This theorem allows us to expand expressions of the form \( (x+y)^n \). In your case, you could let \( x = 5 \) and \( y = -\frac{a}{2} \), and then apply the theorem to expand it in a fun and systematic way! You can also visualize the expression as a cube of a quantity, which can represent geometric volume! If you think of \( 5 - \frac{a}{2} \) as the length of a side of a cube, then cubing that expression gives you the volume of a cube, showcasing how algebra can intersect with three-dimensional thinking!
