Multiply the algebraic expression using a Special Product Formula and simplify.

(1 − 9*r*)3

To multiply the algebraic expression \((1 - 9r)^3\) using a special product formula, we can use the **binomial expansion** formula (also known as the Binomial Theorem). The Binomial Theorem states that for any integers \(n \geq 0\) and any numbers \(a\) and \(b\),

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k.
\]

For the expression \((1 - 9r)^3\), we have \(a = 1\), \(b = -9r\), and \(n = 3\). Applying the binomial theorem, the expansion is:

\[
(1 - 9r)^3 = \sum_{k=0}^{3} \binom{3}{k} (1)^{3-k} (-9r)^k.
\]

Let's compute each term in the sum:

1. **Term for \(k = 0\):**

\[
\binom{3}{0} (1)^{3-0} (-9r)^0 = 1 \cdot 1 \cdot 1 = 1.
\]

2. **Term for \(k = 1\):**

\[
\binom{3}{1} (1)^{3-1} (-9r)^1 = 3 \cdot 1 \cdot (-9r) = -27r.
\]

3. **Term for \(k = 2\):**

\[
\binom{3}{2} (1)^{3-2} (-9r)^2 = 3 \cdot 1 \cdot 81r^2 = 243r^2.
\]

4. **Term for \(k = 3\):**

\[
\binom{3}{3} (1)^{3-3} (-9r)^3 = 1 \cdot 1 \cdot (-729r^3) = -729r^3.
\]

Now, sum all the terms:

\[
(1 - 9r)^3 = 1 - 27r + 243r^2 - 729r^3.
\]

Thus, the simplified expression is:

\[
\boxed{1 - 27r + 243r^2 - 729r^3}.
\]

Would you like further details or have any questions? 

Here are 5 related questions to expand on this topic:

1. How do you use the Binomial Theorem for different values of \(n\)?
2. Can you apply the binomial expansion to expressions with higher powers, such as \((2x - 3y)^4\)?
3. What is the difference between expanding \((a + b)^n\) and \((a - b)^n\)?
4. How do combinations \(\binom{n}{k}\) relate to binomial expansion?
5. What is Pascal's Triangle, and how does it relate to the Binomial Theorem?

**Tip:** Remember, when expanding binomials, the signs in each term alternate if the second term of the binomial is negative.

Learn how to multiply and simplify the algebraic expression (1 - 9r)^3 using the binomial expansion formula. This problem demonstrates the application of the Binomial Theorem in algebraic expressions. Suitable for high school students.

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