Multiply the algebraic expression using a Special Product Formula and simplify(1 − 4*r*)3

To multiply and simplify the algebraic expression \((1 - 4r)^3\) using a special product formula, we can use the **cube of a binomial** formula. The formula is:

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

Here, \(a = 1\) and \(b = 4r\). Now, apply the formula:

\[
(1 - 4r)^3 = 1^3 - 3(1)^2(4r) + 3(1)(4r)^2 - (4r)^3
\]

Let's calculate each term:

1. \(1^3 = 1\)
2. \(-3(1)^2(4r) = -3 \times 1 \times 4r = -12r\)
3. \(3(1)(4r)^2 = 3 \times 1 \times 16r^2 = 48r^2\)
4. \(-(4r)^3 = -64r^3\)

Now, combine all the terms:

\[
(1 - 4r)^3 = 1 - 12r + 48r^2 - 64r^3
\]

So, the simplified expression is:

\[
1 - 12r + 48r^2 - 64r^3
\]

Would you like further explanation, or do you have any questions?

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Here are 5 related questions:
1. How would you expand \((2x + 5)^3\) using the cube of a binomial formula?
2. What is the expansion of \((x - 3y)^3\)?
3. Can you simplify \((a + b)^3\) using the special product formula?
4. How do you recognize when to apply the cube of a binomial formula?
5. What is the difference between \((a + b)^3\) and \((a - b)^3\) in their expanded forms?

**Tip:** When dealing with cubic binomials, always remember the special product formulas—they can save you a lot of time and effort!

Learn how to multiply and simplify the algebraic expression (1 - 4r)^3 using the cube of a binomial formula. This detailed step-by-step solution covers key algebraic concepts, including the cube of a binomial formula and simplification of cubic expressions. Suitable for high school students.

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