a+b whole cube | A Plus B Whole Cube Formula

Edited By Team Careers360 | Updated on Jul 02, 2025 05:32 PM IST

Assume that a and b are two variables that, in algebraic form, stand in for two terms. In mathematics, the sum of the two terms is denoted by the symbol(a+b). It is both a binomial and an algebraic expression. In mathematics, the cube of the sum of the terms a and b, or a binomial, is represented by the following notation:(a+b)3

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Introduction of a+b whole cube
Usage
Examples
Proofs

Introduction of a+b whole cube

When a cube is added with b cube and then the result is added with three times a multiply by b, and addition of a and b, then the a plus b whole cube is obtained.

This can be represented as-

(a + b)3 =a³ + b3 + 3ab(a+b)

The formula for a plus b whole cube can also be written as a cube plus b cube plus three a square b plus three a, b square. It is written below-

(a + b)3 = a3+ b3 + 3a2b+3ab2

In mathematics, the plus b whole cubed algebraic identity is called in the following three ways-

The cube of the sum of two terms rules.
The cube of a binomial identity.
The special binomial product formula.

Usage

In the following two instances, the cube of the sum of two terms rule is applied as a formula.

1. Expansion

The total of the cubes of the two terms and their product, multiplied by three, form the cube of the sum of the two terms.

(a + b)3 = a3+ b3 + 3ab(a+b)

2.Simplification

The cube of the sum of two terms is the total of the cubes of the two terms plus three times the product of the two terms added together. a3+ b3 + 3ab(a+b)=(a + b)3

Examples

Example 1 - Solve (2 + 3)3

Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3

So here a = 2 and b = 3

So we will put these values in the formula, after putting the values we will get-

23+ 33 + 3*2*3*(2+3)=(2 + 3)3

= 8+27+18 *5

=8+27+90

=125

Hence, 125 is the answer.

Example 2 - Solve (1e + 2d)3

Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3

So here a = 1e and b = 2d

So we will put these values in the formula, after putting the values we will get-

(1e)3+ (2d)3 + 3*1e*2d*(1e+2d)=(1e + 2d)3

= e3+8d3+6ed *(1e+2d)

=e3+8d3+6e2d+12ed2

Hence, e3+8d3+6e2d+12ed2 is the answer.

Example 3 - Solve (3h + i)3

Answer- We know the formula a3+ b3 + 3ab(a+b)=(a + b)3

So here a = 3h and b = i

So we will put these values in the formula, after putting the values we will get-

(3h)3+ i3 + 3*3h*i*(3h+i)=(3h + i)3

= 27h3+i3+9hi *(3h+i)

=27h3+i3+27h2i+9hi2

Hence , 27h3+i3+27h2i+9hi2 is the answer.

Proofs

The following two distinct mathematical techniques can be used to demonstrate the algebraic identity of a plus b for the full cube.

1. Algebraic approach

It is beneficial to use the product of three sum basis binomials to get the expansion of the a plus b whole cube formula.

2. Geometric approach

A cube's volume can be used to graphically demonstrate how the algebraic identity for a plus b entire cube can be expanded.

Frequently Asked Questions (FAQs)

1. What is a plus b whole cube formula?

When a cube is added with b cube and then the result is added with three times a multiplied by b , and addition of a and b , then the a plus b whole cube is obtained.

This can be represented as-

(a + b)3 = a3+ b3 + 3ab(a+b)

2. Which mathematical techniques can be used to demonstrate the algebraic identity of a plus b for the full cube?

The following two distinct mathematical techniques can be used to demonstrate the algebraic identity of a plus b for the full cube.

1. Algebraic approach

It is beneficial to use the product of three sum basis binomials to get the expansion of the a plus b whole cube formula.

2. Geometric approach

A cube's volume can be used to graphically demonstrate how the algebraic identity for a plus b entire cube can be expanded.

3. What are the applications of a plus b whole cube formula?

The applications of a plus b whole cube formula are given below. It can be used in-

1. Expansion

The total of the cubes of the two terms and their product, multiplied by three, form the cube of the sum of the two terms.

(a + b)3 = a3+ b3 + 3ab(a+b)

2.Simplification

The cube of the sum of two terms is the total of the cubes of the two terms plus three times the product of the two terms added together. a3+ b3 + 3ab(a+b)=(a + b)3

4. What are the other names for a plus b whole cubed algebraic identity ?

The other names for a plus b whole cubed algebraic identity are-

The cube of the sum of two terms rules.
The cube of a binomial identity.
The special binomial product formula.

5. Write the formula for a plus b whole cube.

The formula for a plus b whole cube is -

(a + b)3 = a3+ b3 + 3ab(a+b)

It can also be represented as-

(a + b)3 = a3+ b3 + 3a2b+3ab2

6. How does the a+b whole cube formula relate to the concept of perfect cubes?

The a+b whole cube formula expands a perfect cube trinomial. If you can factor an expression into the form of this formula, then the original expression is a perfect cube and can be written as (a+b)³.

7. Can the a+b whole cube formula be used for factoring?

Yes, the formula can be used in reverse for factoring. If you encounter an expression of the form a³ + 3a²b + 3ab² + b³, you can factor it as (a+b)³.

8. What's the geometric interpretation of the a+b whole cube formula?

Geometrically, (a+b)³ represents the volume of a cube with side length (a+b). The individual terms represent smaller volumes: a³ (a large cube), 3a²b (three rectangular prisms), 3ab² (three more rectangular prisms), and b³ (a small cube).

9. How does the a+b whole cube formula change if a or b are negative?

The formula remains the same, but you need to be careful with signs. For example, (-a+b)³ = (-a)³ + 3(-a)²b + 3(-a)b² + b³ = -a³ + 3a²b - 3ab² + b³.

10. How can you check if you've correctly applied the a+b whole cube formula?

You can check your work by expanding (a+b)(a+b)(a+b) manually and comparing the result to your application of the formula. Alternatively, you can substitute numerical values for a and b and verify that both sides of the equation are equal.

11. What is the a+b whole cube formula?

The a+b whole cube formula, also known as the cube of a binomial, is (a+b)³ = a³ + 3a²b + 3ab² + b³. This formula expands the cube of the sum of two terms, showing all the terms that result from this operation.

12. How does the a+b whole cube formula relate to the concept of algebraic identities?

The a+b whole cube formula is an important algebraic identity. It's part of a family of identities including the square of a binomial (a+b)² and the difference of cubes a³-b³. These identities are fundamental tools in algebra for expanding and factoring expressions.

13. How is the a+b whole cube formula different from the square of a binomial?

The a+b whole cube formula expands (a+b)³, while the square of a binomial expands (a+b)². The cube formula has four terms: a³, 3a²b, 3ab², and b³, whereas the square formula has three terms: a², 2ab, and b².

14. What's the difference between (a+b)³ and a³+b³?

(a+b)³ expands to a³ + 3a²b + 3ab² + b³, while a³+b³ is just the sum of the cubes of a and b. The difference between these expressions, 3a²b + 3ab², is called the "middle terms" of the cube expansion.

15. How is the a+b whole cube formula related to the binomial theorem?

The a+b whole cube formula is a specific case of the binomial theorem where n=3. The binomial theorem provides a general formula for expanding (a+b)ⁿ for any positive integer n.

16. Why does the a+b whole cube formula have four terms?

The four terms in the a+b whole cube formula represent all possible combinations of a and b when multiplied three times. Each term corresponds to choosing a or b in each of the three factors: (a+b)(a+b)(a+b).

17. What is the significance of the coefficients in the a+b whole cube formula?

The coefficients (1, 3, 3, 1) in the formula represent the number of ways to choose a or b in each term. For example, there are 3 ways to choose two a's and one b, hence the coefficient 3 in the term 3a²b.

18. How can you derive the a+b whole cube formula?

You can derive the formula by multiplying (a+b) by itself three times: (a+b)(a+b)(a+b). Alternatively, you can use the binomial theorem with n=3, which gives the same result.

19. What is the relationship between the a+b whole cube formula and Pascal's triangle?

The coefficients in the a+b whole cube formula (1, 3, 3, 1) correspond to the fourth row of Pascal's triangle. This relationship holds for higher powers as well, with each row representing coefficients for successive powers.

20. What happens if you substitute -b for b in the a+b whole cube formula?

If you replace b with -b, you get the a-b whole cube formula: (a-b)³ = a³ - 3a²b + 3ab² - b³. Notice that the signs of the terms alternate, but the coefficients remain the same.

21. How does the a+b whole cube formula relate to volume expansion?

The a+b whole cube formula can be visualized as the volume expansion of a cube with side length (a+b). Each term represents a part of the expanded volume: a³ is a large cube, 3a²b and 3ab² are rectangular prisms, and b³ is a small cube.

22. How does the a+b whole cube formula relate to polynomial long division?

The a+b whole cube formula can be useful in polynomial long division when the dividend is a perfect cube. Recognizing the expanded form can help in identifying factors and simplifying the division process.

23. Can the a+b whole cube formula be applied to algebraic expressions with more than two terms?

No, the a+b whole cube formula is specifically for binomials (two terms). For expressions with more than two terms, you would need to use more complex multinomial expansions or the general binomial theorem.

24. Why are there no squared terms in the a+b whole cube formula?

While there are no explicit squared terms, a² and b² are present within the formula. They appear in the middle terms 3a²b and 3ab². The absence of pure squared terms reflects the fact that we're dealing with a cube, not a square.

25. Can the a+b whole cube formula be applied to complex numbers?

Yes, the formula works for complex numbers as well. You would apply it in the same way, but you need to be careful with the arithmetic of complex numbers when expanding each term.

26. Why is it important to memorize the a+b whole cube formula?

While memorization can be helpful, it's more important to understand the concept behind the formula. Understanding how it's derived and what each term represents allows you to reconstruct it if forgotten and apply it more flexibly to problem-solving.

27. How can you use the a+b whole cube formula to calculate (101)³?

To calculate (101)³, we can write 101 as (100+1) and apply the formula: (100+1)³ = 100³ + 3(100²)(1) + 3(100)(1²) + 1³ = 1,000,000 + 30,000 + 300 + 1 = 1,030,301.

28. What's the connection between the a+b whole cube formula and combinatorics?

The coefficients in the formula (1, 3, 3, 1) represent the number of ways to choose 0, 1, 2, or 3 b's from three factors. This relates to combinations in combinatorics, specifically ³C₀, ³C₁, ³C₂, and ³C₃.

29. How does the a+b whole cube formula extend to higher powers?

For higher powers, you would use the general binomial theorem. The pattern of terms and coefficients becomes more complex, but the principle remains the same. For example, (a+b)⁴ would have five terms with coefficients 1, 4, 6, 4, 1.

30. What's the relationship between the a+b whole cube formula and the a-b whole cube formula?

The a-b whole cube formula can be derived from the a+b formula by replacing b with -b. This changes the signs of the odd-power terms: (a-b)³ = a³ - 3a²b + 3ab² - b³.

31. How can you use the a+b whole cube formula to approximate cube roots?

The formula can be used to approximate cube roots by writing a number as (a+b)³ and solving for b. For example, to find the cube root of 28, you might write 28 as (3+b)³ and solve the resulting equation.

32. What role does the a+b whole cube formula play in calculus?

In calculus, the formula is useful for expanding functions before differentiation or integration. It's also used in Taylor series expansions, where it represents the third-degree term in the series for (1+x)³.

33. How does the a+b whole cube formula relate to the concept of polynomial functions?

The expanded form of (a+b)³ is a third-degree polynomial function. Understanding this formula helps in analyzing the behavior of cubic functions, including their graphs and roots.

34. Can the a+b whole cube formula be generalized to more than two variables?

Yes, there are multinomial expansions for (a+b+c)³ and higher. These formulas are more complex but follow similar principles. For example, (a+b+c)³ expands to 10 terms instead of 4.

35. How does the a+b whole cube formula relate to the concept of perfect cube trinomials?

A perfect cube trinomial is an expression that can be factored into the form (a+b)³. The a+b whole cube formula gives the expanded form of such trinomials, making it easier to recognize and factor them.

36. What's the significance of the middle terms in the a+b whole cube formula?

The middle terms, 3a²b and 3ab², are crucial for understanding the full expansion. They represent the interaction between a and b when cubed and are often the terms that students forget or miscalculate.

37. How can you use the a+b whole cube formula to solve cubic equations?

While the formula itself doesn't directly solve cubic equations, recognizing when a cubic equation is in the form of (x+k)³ = m can simplify the solving process, as you can then take the cube root of both sides.

38. What's the connection between the a+b whole cube formula and the sum and difference of cubes formulas?

The a+b whole cube formula expands (a+b)³, while the sum of cubes formula factors a³+b³. They're related but inverse operations. The difference of cubes formula factors a³-b³, which is related to (a-b)³.

39. How does understanding the a+b whole cube formula help in factoring polynomials?

Recognizing the expanded form of (a+b)³ in a polynomial can help identify perfect cube factors. This knowledge can simplify the factoring process for certain types of polynomials.

40. What's the importance of the a+b whole cube formula in number theory?

In number theory, the formula is useful for understanding properties of cubic numbers and for certain types of Diophantine equations. It's also related to concepts like cubic residues.

41. How can you use the a+b whole cube formula to mentally calculate cubes of numbers close to 100?

For numbers close to 100, you can write them as 100±x and use the formula. For example, 103³ = (100+3)³ = 1,000,000 + 3(10,000) + 3(300) + 27 = 1,092,727.

42. What's the relationship between the a+b whole cube formula and the concept of polynomial expansion?

The a+b whole cube formula is a specific case of polynomial expansion. It demonstrates how terms interact when raised to a power, which is a fundamental concept in understanding more general polynomial expansions.

43. How does the a+b whole cube formula relate to the concept of algebraic proofs?

The formula can be used in algebraic proofs, especially those involving cubic expressions. Understanding how to derive and apply this formula is crucial for constructing and following such proofs.

44. What's the significance of the a+b whole cube formula in the study of polynomial functions?

The formula provides insights into the behavior of cubic functions. It helps in understanding how changing the coefficients affects the graph and roots of these functions.

45. How can you use the a+b whole cube formula to understand the concept of polynomial identities?

The a+b whole cube formula is itself a polynomial identity. By studying how it's derived and applied, students can better understand the nature of polynomial identities in general.

46. What's the connection between the a+b whole cube formula and the concept of algebraic expansion?

The a+b whole cube formula is a prime example of algebraic expansion. It demonstrates how complex expressions can be derived from simpler ones through systematic multiplication and combination of terms.

47. How does the a+b whole cube formula relate to the study of polynomial equations?

Understanding this formula is crucial for solving certain types of cubic equations. It also provides insights into the nature of cubic polynomials and their roots.

48. What role does the a+b whole cube formula play in understanding algebraic structures?

The formula demonstrates important properties of algebraic structures, such as the distributive property and how exponents behave. It's a concrete example of how complex algebraic expressions can be built from simpler ones.

49. How can the a+b whole cube formula be used to understand the concept of algebraic manipulation?

The formula provides a clear example of how algebraic terms can be combined and simplified. Understanding how to derive and use this formula helps develop skills in algebraic manipulation.

50. What's the significance of the a+b whole cube formula in the context of mathematical modeling?

In mathematical modeling, the formula can be used to describe cubic growth or decay processes. It's particularly useful when modeling phenomena that involve volume or three-dimensional growth.

51. How does the a+b whole cube formula relate to the concept of algebraic factoring?

The formula can be used in reverse for factoring. Recognizing when a polynomial is in the expanded form of (a+b)³ allows for quick factorization, which is a key skill in algebra.

52. What's the connection between the a+b whole cube formula and the study of polynomial functions?

The formula provides insights into the behavior of cubic functions, including their graphs and roots. Understanding this formula helps in analyzing and predicting the characteristics of these functions.

53. How can the a+b whole cube formula be used to understand the concept of algebraic generalization?

The formula is a specific case of the more general binomial theorem. By understanding this formula, students can begin to grasp how algebraic concepts can be generalized to more complex situations.

54. What's the importance of the a+b whole cube formula in developing algebraic thinking?

Working with this formula helps develop algebraic thinking by requiring students to understand patterns, generalize from specific cases, and manipulate symbolic representations.

55. How does the a+b whole cube formula contribute to the broader understanding of algebra and its applications?

The formula serves as a bridge between basic algebraic concepts and more advanced topics. It demonstrates how simple ideas (like addition and multiplication) can be combined to create powerful tools for solving complex problems in mathematics and real-world applications.