To expand and simplify the expression [tex]\(\left(c + \frac{d}{e}\right)^3\)[/tex], follow these steps:
1. Understand the Expression:
The expression given is of the form [tex]\((a + b)^3\)[/tex], where [tex]\(a = c\)[/tex] and [tex]\(b = \frac{d}{e}\)[/tex].
2. Use the Binomial Theorem:
Recall the binomial theorem for expanding [tex]\((a + b)^3\)[/tex]:
[tex]\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\][/tex]
3. Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Here, substitute [tex]\(a = c\)[/tex] and [tex]\(b = \frac{d}{e}\)[/tex]:
[tex]\[
\left(c + \frac{d}{e}\right)^3 = c^3 + 3c^2\left(\frac{d}{e}\right) + 3c\left(\frac{d}{e}\right)^2 + \left(\frac{d}{e}\right)^3
\][/tex]
4. Simplify Each Term:
- The first term is simply:
[tex]\[
c^3
\][/tex]
- The second term, [tex]\(3c^2\left(\frac{d}{e}\right)\)[/tex], simplifies to:
[tex]\[
3c^2 \cdot \frac{d}{e} = \frac{3c^2d}{e}
\][/tex]
- The third term, [tex]\(3c\left(\frac{d}{e}\right)^2\)[/tex], simplifies to:
[tex]\[
3c \cdot \left(\frac{d}{e}\right)^2 = 3c \cdot \frac{d^2}{e^2} = \frac{3cd^2}{e^2}
\][/tex]
- The fourth term, [tex]\(\left(\frac{d}{e}\right)^3\)[/tex], simplifies to:
[tex]\[
\left(\frac{d}{e}\right)^3 = \frac{d^3}{e^3}
\][/tex]
5. Combine All Terms:
Now, combine all the simplified terms together:
[tex]\[
\left(c + \frac{d}{e}\right)^3 = c^3 + \frac{3c^2d}{e} + \frac{3cd^2}{e^2} + \frac{d^3}{e^3}
\][/tex]
Hence, the fully expanded form of [tex]\(\left(c + \frac{d}{e}\right)^3\)[/tex] is:
[tex]\[
c^3 + \frac{3c^2d}{e} + \frac{3cd^2}{e^2} + \frac{d^3}{e^3}
\][/tex]
So the final answer is:
[tex]\[
c^3 + 3\frac{c^2d}{e} + 3\frac{cd^2}{e^2} + \frac{d^3}{e^3}
\][/tex]
