What is (a-b) Cubed?

S3-SA1-0754

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The expression "(a-b) cubed" means multiplying (a-b) by itself three times. It's written as (a-b)^3. When expanded, it follows a special algebraic identity: a^3 - 3a^2b + 3ab^2 - b^3.

Simple Example

Quick Example

Imagine you have 'a' mangoes and give 'b' mangoes away. You are left with (a-b) mangoes. If you want to know the 'cube' of this remaining amount, you'd calculate (a-b)^3. For example, if you start with 10 mangoes (a=10) and give 2 away (b=2), you have 8 mangoes left. (10-2)^3 is 8^3, which is 512.

Worked Example

Step-by-Step

Let's expand (x-3)^3 using the identity (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
---Step 1: Identify 'a' and 'b'. Here, a = x and b = 3.
---Step 2: Substitute 'a' and 'b' into the first term, a^3. This becomes x^3.
---Step 3: Substitute 'a' and 'b' into the second term, -3a^2b. This becomes -3(x^2)(3) = -9x^2.
---Step 4: Substitute 'a' and 'b' into the third term, +3ab^2. This becomes +3(x)(3^2) = +3(x)(9) = +27x.
---Step 5: Substitute 'a' and 'b' into the fourth term, -b^3. This becomes -(3^3) = -27.
---Step 6: Combine all the terms. So, (x-3)^3 = x^3 - 9x^2 + 27x - 27.
---Answer: x^3 - 9x^2 + 27x - 27

Why It Matters

Understanding algebraic identities like (a-b)^3 helps simplify complex calculations in fields like engineering and physics. It's used by software developers to write efficient code and by data scientists to model relationships between different data points. Even in game development, these formulas help calculate object movements!

Common Mistakes

MISTAKE: Writing (a-b)^3 as a^3 - b^3 | CORRECTION: Remember the full identity: (a-b)^3 has four terms, not just two. It's a^3 - 3a^2b + 3ab^2 - b^3.

MISTAKE: Forgetting the alternating signs, especially in the middle terms | CORRECTION: The signs for (a-b)^3 are always +, -, +, -. For (a+b)^3, all signs are +.

MISTAKE: Incorrectly calculating 3a^2b or 3ab^2, especially squaring the wrong term | CORRECTION: Always square 'a' in the second term (a^2) and 'b' in the third term (b^2). Double-check your multiplication.

Practice Questions

Try It Yourself

QUESTION: Expand (y-2)^3. | ANSWER: y^3 - 6y^2 + 12y - 8

QUESTION: If (x-y)^3 = x^3 - 3x^2y + Pxy^2 - y^3, what is the value of P? | ANSWER: P = 3

QUESTION: A cube has a side length of (5k-1) cm. Find the expression for its volume. | ANSWER: 125k^3 - 75k^2 + 15k - 1 cm^3

MCQ

Quick Quiz

Which of the following is the correct expansion of (2p-1)^3?

8p^3 - 1

8p^3 - 12p^2 + 6p - 1

8p^3 + 12p^2 + 6p + 1

8p^3 - 6p^2 + 12p - 1

The Correct Answer Is:

Using the identity (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3, with a=2p and b=1, we get (2p)^3 - 3(2p)^2(1) + 3(2p)(1)^2 - (1)^3, which simplifies to 8p^3 - 12p^2 + 6p - 1.

Real World Connection

In the Real World

Engineers designing bridges or buildings use similar algebraic expressions to calculate how materials behave under stress or how much space a structure will occupy. For instance, when designing a new metro station, they might use these formulas to calculate the volume of concrete needed for a part of the structure, even if its dimensions are expressed algebraically.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables | CUBE: Multiplying a number or expression by itself three times | EXPAND: To multiply out the terms in an expression | TERM: A single number, variable, or product of numbers and variables in an expression | VARIABLE: A letter representing an unknown number

What's Next

What to Learn Next

Great job learning about (a-b) cubed! Next, you can explore "What is (a+b) Cubed?" This will help you see the similarities and differences between these two important identities and make you even stronger in algebra!