What is (a+b) Cubed?

S3-SA1-0753

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: (a+b) x (a+b) x (a+b). It expands into a formula that helps us quickly calculate this value without doing all the multiplications separately.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, and you want to know how many ways they can arrange themselves in three different spots for a photo. If 'a' represents Rohan and 'b' represents Priya, (a+b) cubed is like figuring out all the combinations if they could appear multiple times. The formula helps us count these possibilities easily.

Worked Example

Step-by-Step

Let's find the value of (2+3) Cubed using the formula (a+b)^3 = a^3 + b^3 + 3ab(a+b).
---Step 1: Identify 'a' and 'b'. Here, a = 2 and b = 3.
---Step 2: Calculate a^3. a^3 = 2^3 = 2 x 2 x 2 = 8.
---Step 3: Calculate b^3. b^3 = 3^3 = 3 x 3 x 3 = 27.
---Step 4: Calculate 3ab. 3ab = 3 x 2 x 3 = 18.
---Step 5: Calculate (a+b). (a+b) = 2 + 3 = 5.
---Step 6: Substitute these values into the formula: 8 + 27 + 18(5).
---Step 7: Continue the calculation: 8 + 27 + 90.
---Step 8: Add the numbers: 35 + 90 = 125.
So, (2+3) Cubed = 125.

Why It Matters

Understanding (a+b) Cubed helps in fields like engineering to design structures efficiently, in computer science for writing algorithms, and even in economics to model growth. Learning this foundational concept now opens doors to exciting careers as an AI engineer, data scientist, or even a physicist at ISRO.

Common Mistakes

MISTAKE: Students often write (a+b)^3 as a^3 + b^3 | CORRECTION: Remember the formula includes the 3ab(a+b) term. It's not just the sum of cubes!

MISTAKE: Incorrectly calculating 3ab(a+b) by forgetting to multiply by (a+b) | CORRECTION: Always calculate 3ab first, then multiply that result by the sum of 'a' and 'b'.

MISTAKE: Forgetting that 'cubed' means multiplying a number by itself three times, not by 3 | CORRECTION: a^3 means a x a x a, not a x 3.

Practice Questions

Try It Yourself

QUESTION: Expand (x+y)^3. | ANSWER: x^3 + y^3 + 3xy(x+y)

QUESTION: If a=1 and b=4, what is the value of (a+b)^3? | ANSWER: 125

QUESTION: Find the value of (10+2)^3 using the formula. | ANSWER: 1728

MCQ

Quick Quiz

Which of the following is the correct expansion of (p+q) Cubed?

p^3 + q^3

p^3 + q^3 + 3pq(p+q)

p^3 + q^3 + 3pq

3pq(p+q)

The Correct Answer Is:

The correct formula for (p+q) Cubed is p^3 + q^3 + 3pq(p+q). Options A, C, and D are incomplete or incorrect expansions of the formula.

Real World Connection

In the Real World

Imagine an architect designing a new building in Mumbai. They might use concepts like (a+b) Cubed to calculate the volume of complex shapes or to estimate the amount of materials needed, like cement or steel. This helps them plan efficiently and avoid waste, just like optimizing delivery routes for Zepto.

Key Vocabulary

Key Terms

CUBED: A number multiplied by itself three times, like 2^3 = 2x2x2 | EXPANSION: Writing out an expression in a longer form by performing multiplications | ALGEBRAIC IDENTITY: An equation that is true for all possible values of its variables | TERM: A single number or variable, or numbers and variables multiplied together, separated by + or - signs.

What's Next

What to Learn Next

Great job understanding (a+b) Cubed! Next, you should explore (a-b) Cubed. It uses a very similar idea but with some key differences in the signs, which will further strengthen your algebraic skills.