Multiply the algebraic expression using a Special Cube 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 **cube of a binomial** formula. The cube of a binomial \((a - b)^3\) is given by:

\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
\]

For the expression \((1 - 9r)^3\), let \(a = 1\) and \(b = 9r\). Substituting these into the formula:

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

Now, calculate each term:

1. **First term: \(a^3 = 1^3\):**

\[
1^3 = 1.
\]

2. **Second term: \(-3a^2b = -3(1)^2(9r)\):**

\[
-3 \cdot 1 \cdot 9r = -27r.
\]

3. **Third term: \(3ab^2 = 3(1)(9r)^2\):**

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

4. **Fourth term: \(-b^3 = -(9r)^3\):**

\[
-(9r)^3 = -729r^3.
\]

Combine all these terms to simplify the expression:

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

Therefore, the simplified form of the expression is:

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

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you use the cube of a binomial formula for different expressions?
2. Can you explain the difference between the cube of a sum and the cube of a difference?
3. How does the cube of a binomial formula extend to higher powers, like the fourth power?
4. What are other special product formulas, and when should they be used?
5. How do you recognize which special product formula to apply to a given algebraic expression?

**Tip:** The cube of a binomial formula helps quickly expand expressions without individually multiplying each term, saving time and reducing errors.

Learn how to multiply and simplify the algebraic expression (1 - 9r)^3 using the cube of binomial formula step-by-step. This problem explores the application of algebraic concepts, including the cube of binomial theorem. Suitable for high school students.

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