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(a + b)³ Formula | Expansion of a Plus b Whole Cube Identity
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In math, the formula (a + b)³ = a³ + 3a²b + 3ab² + b³ is called an algebraic identity.
It helps us expand the cube of a binomial — that means finding the result when we multiply (a + b) three times.
This identity is useful in solving problems where you need to:
- Expand expressions like (a + b)³, and
- Sometimes factor expressions made of three terms.
So, the formula gives the cube of the sum of two numbers or variables — "a" and "b".
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a + b whole cube Formula
To find the cube of a binomial expression, we often use the (a + b)³ formula. This identity makes it easy to expand expressions like (a + b)³ without doing long multiplication.
It also helps in factoring trinomial expressions in some math problems. Using this formula saves time and avoids complex calculations when you're adding and then cubing two terms.
The formula of
What is a + b whole cube Formula?
We know that the a + b whole cube formula is as follows,
But the bigger question is how do we arrive at the result? Because in mathematics, until we find its derivation it is invalid and of no use to us.
So, now let's work on its derivation.
Clearly, we write,
=>
=>
=>
=>
or,
=>
Hence, we arrive at our result.
Summary of a + b whole cube
Let's summarise the whole discussion so that you can revise quickly and efficiently.
- To calculate the cube of any binomial expression, we generally use the (a + b) whole cube formula.
- (a-b) whole cube formula can be derived from the identity
. - The formula of
can be expressed as,
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a + b whole cube Solved Examples
Example 1: Solve (3x + 4y)³
Solution: We know the identity: (a + b)³ = a³ + 3a²b + 3ab² + b³
Here, let
a = 3x
b = 4y
Apply the formula: (3x + 4y)³ = (3x)³ + 3(3x)²(4y) + 3(3x)(4y)² + (4y)³
Step‑by‑step:
(3x)³ = 27x³
3(3x)²(4y) = 3 × 9x² × 4y = 108x²y
3(3x)(4y)² = 3 × 3x × 16y² = 144xy²
(4y)³ = 64y³
Now add all terms: (3x + 4y)³ = 27x³ + 108x²y + 144xy² + 64y³
Answer: (3x + 4y)³ = 27x³ + 108x²y + 144xy² + 64y³
Example 2: Evaluate
Solution 2: We are given with,
(a + b) = 2 & ab = 1, and
Clearly, we know that,
=>
On Substituting the values of (a + b) and ab in the above expression, we obtained,
=>
=>
=>
=>
Example 3: Can you derive
Solution 3: Yes, we can surely derive (a - b) whole cube formula using the identity
=>
=> \((a + (-b))^3 = (a - b)^3 = (a^3 + 3a^2(-b) + 3a(-b)^2 + (-b)^3)
=>
Hence we obtained,
=>
You can also use it directly as an identity.
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FAQs For a + b whole cube
What is the identity of a+ b whole cube?
The identity (a + b) whole cube equals to
What is the formula of A cube + B cube?
First Formula: a³ + b³ = (a + b)³ − 3a²b − 3ab² Second Formula: a³ + b³ = (a + b)(a² + b² − ab)
What are the different formulas of (a +b)^3?
(a + b)³ = a³ + 3a²b + 3ab² + b³ (a + b)³ = a³ + 3ab(a + b) + b³
What is the binomial expression in Algebra?
In algebra, a binomial expression refers to an algebraic expression that consists of exactly two terms connected by either addition or subtraction. The term "binomial" comes from the Latin words "bi" meaning "two" and "nomial" meaning "term."
What is the Formula of a Plus b Plus c Whole Cube?
Formula for (a + b + c)³ (a plus b plus c whole cube): (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3ab² + 3a²c + 3ac² + 3b²c + 3bc² + 6abc
Can this identity be used for numbers?
Yes! For example, (2 + 3)³ = 125. Using the formula: 2³ + 3×2²×3 + 3×2×3² + 3³ = 8 + 36 + 54 + 27 = 125
How is this identity useful in real-life problems?
It is used in simplifying complex expressions in areas like physics, engineering, computer science, and even in calculating volumes or interest in finance.