a+b whole cube – Algebra Formulas, Get All Algebraic Identities Formulas PDF, Chart, List here

Ranjay Kumar

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Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between quantities. One of the essential algebraic formulas is the expansion of (a+b)3(a + b)^3. This formula, known as the cube of a binomial, is expressed as (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Understanding and applying this identity is crucial for solving complex algebraic problems efficiently.

For students and professionals who seek a comprehensive understanding of algebraic identities, having a well-organized list or chart is invaluable. These resources not only provide quick references but also aid in better grasping the concepts by offering visual representations. To cater to this need, we have compiled a detailed PDF encompassing all major algebraic identities. This includes formulas like the square of a binomial, the difference of squares, and the sum and difference of cubes, among others.

Our downloadable PDF chart serves as a handy tool for anyone looking to master algebra. Whether you are a student preparing for exams, a teacher creating lesson plans, or a professional brushing up on your skills, this comprehensive list of algebraic identities will be an indispensable resource in your mathematical toolkit.

Algebra is a branch of mathematics that uses letters to represent numbers. In an algebraic equation, the operations performed on one side of the equation must also be performed on the other side to maintain balance. While mathematical numbers are constants, algebra also encompasses real numbers, complex numbers, matrices, vectors, and more. Commonly used variables in algebraic problems and equations include X, Y, A, and B.

Algebra Formulas

Algebra formulas form the foundation for many critical topics in mathematics. Equations, quadratic equations, polynomials, coordinate geometry, calculus, trigonometry, and probability all rely heavily on algebraic formulas to understand and solve complex problems.

Definition of Algebra Formulas

Algebra formulas are a vital part of Class 10 mathematics in India. Algebra is one of the most crucial areas of mathematics, encompassing numerous topics such as quadratic equations, polynomials, coordinate geometry, calculus, trigonometry, probability, and more. These formulas use a combination of numbers and letters, with common variables being X, Y, A, and B. Algebraic formulas allow us to solve time-consuming algebraic problems quickly and efficiently. Here, we provide all significant algebraic formulas along with their solutions, making them accessible to students in one place.

Algebra formulas are essentially algebraic equations composed of mathematical phrases and symbols. These formulas typically include an unknown variable xx, which is revealed during the simplification of an equation. Algebraic formulas are useful for solving complex algebraic computations in a straightforward manner.

For example:

(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

In the formula above, both sides are algebraic expressions. The expression a3+3a2b+3ab2+b3a^3 + 3a^2b + 3ab^2 + b^3 is the simplified form of (a+b)3(a + b)^3.

Algebraic Formulas and Identities

In algebra, an identity is an equation that holds true for all values of the variables involved. An algebraic identity means that the left-hand side (LHS) of the equation is exactly equal to the right-hand side (RHS) for any value of the variables. These identities are particularly useful for solving equations and simplifying expressions. Here are some commonly used algebraic identities:

Algebraic Identities Formulas

  • (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • (a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2
  • (a+b)(a−b)=a2−b2(a + b)(a – b) = a^2 – b^2
  • (x+a)(x+b)=x2+x(a+b)+ab(x + a)(x + b) = x^2 + x(a + b) + ab

Algebra Formulas for Squares (Class 10)

Here are some important formulas involving squares:

  • a2−b2=(a−b)(a+b)a^2 – b^2 = (a – b)(a + b)
  • (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 – 2ab
  • (a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2
  • (a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
  • (a−b−c)2=a2+b2+c2−2ab+2bc−2ca(a – b – c)^2 = a^2 + b^2 + c^2 – 2ab + 2bc – 2ca

These formulas are fundamental in algebra and are essential tools for solving various mathematical problems.

Algebra Formulas for Cube of SSC CGL

Here are some Algebraic formulas involving cubes.

• (a + b)³ = a³+ 3a²b + 3ab²+ b³
•  (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³= a³ – 3a²b + 3ab² – b³
• (a – b)³= a³ – b³ – 3ab(a – b)
• a³ – b³ = (a – b)(a²+ ab + b²)
• a³ + b³ = (a + b)(a²– ab + b²)

Some more Algebra formulas are –
• (a + b)⁴= a⁴+ 4a³b + 6a²b² + 4ab³ + b²
• (a – b)⁴= a4 – 4a³b + 6a²b² – 4ab³+ b⁴
• a⁴ – b⁴= (a – b)(a + b)(a² + b²)
• a⁵ – b⁵= (a – b)(a⁴ + a³b + a²b² + ab³+ b⁴)

Algebra Formulas for Natural Numbers

Natural numbers are the positive integers starting from 1 and increasing without bound (i.e., from 1 to infinity). These numbers exclude 0 and negative numbers. Several algebraic formulas are commonly used when performing operations on natural numbers. Let’s consider nn to be a natural number.

Key Formulas

  1. Difference of Powers:

    an−bn=(a−b)(an−1+an−2b+⋯+abn−2+bn−1)a^n – b^n = (a – b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})
  2. Sum of Powers (for even nn):

    an+bn=(a+b)(an−1−an−2b+⋯+abn−2−bn−1)where n is evena^n + b^n = (a + b)(a^{n-1} – a^{n-2}b + \cdots + ab^{n-2} – b^{n-1}) \quad \text{where } n \text{ is even}
  3. Sum of Powers (for odd nn):

    an+bn=(a+b)(an−1−an−2b+an−3b2−⋯−bn−2a+bn−1)where n is odda^n + b^n = (a + b)(a^{n-1} – a^{n-2}b + a^{n-3}b^2 – \cdots – b^{n-2}a + b^{n-1}) \quad \text{where } n \text{ is odd}

Laws of Exponents

In algebra, an exponent (or power) represents repeated multiplication of a number. For example, 3×3×3×33 \times 3 \times 3 \times 3 can be written as 343^4, where 4 is the exponent of 3. Exponents indicate how many times a number is multiplied by itself. The rules for operating with exponents in addition, subtraction, and multiplication can be easily solved using algebraic formulas.

Basic Rules of Exponents

  1. Product of Powers Rule:

    am⋅an=am+na^m \cdot a^n = a^{m+n}
  2. Quotient of Powers Rule:

    aman=am−n(for m>n)\frac{a^m}{a^n} = a^{m-n} \quad \text{(for } m > n\text{)}
  3. Power of a Power Rule:

    (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  4. Power of a Product Rule:

    (ab)n=an⋅bn(ab)^n = a^n \cdot b^n
  5. Power of a Quotient Rule:

    (ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

These formulas and rules are fundamental in algebra and are widely used for simplifying and solving equations involving natural numbers.

Algebra Formulas for Quadratic Equations

Algebra formulas for quadratic equations are crucial topics in the Class 9 and 10 syllabus. To find the roots of a given quadratic equation, we use the following formulas:a+b whole cube

If ax2+bx+c=0ax^2 + bx + c = 0 is a quadratic equation, then:

Based on the given formula, we can conclude the following if the roots of the quadratic equation are α and β:a+b whole cube

  1. The equation will be (x−α)(x−β)=0(x – \alpha)(x – \beta) = 0.
  2. The sum of the roots (α+β)(\alpha + β) is equal to −ba-\frac{b}{a}, and the product of the roots (α×β)(\alpha \times β) is equal to ca\frac{c}{a}.
 

Algebra Formulas for Irrational Numbers (SSC CGL)

Here are the key formulas used to solve equations involving irrational numbers:

  1. ab=a⋅b\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}
  2. ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
  3. (a+b)(a−b)=a−b(\sqrt{a} + \sqrt{b})(\sqrt{a} – \sqrt{b}) = a – b
  4. (a+b)2=a+2ab+b(\sqrt{a} + \sqrt{b})^2 = a + 2\sqrt{ab} + b
  5. (a+b)(a−b)=a2−b(a + \sqrt{b})(a – \sqrt{b}) = a^2 – b

Algebra Formulas List and Sheet

Here is a comprehensive list of all important algebraic formulas. Students should review this list to solve complex algebraic equations quickly and efficiently.

Important Formulas 
 1a²– b² = (a – b)(a + b)
2(a + b)²=  a²+ 2ab + b²
3a²+ b²= (a + b)²– 2ab
4(a – b)² = a²– 2ab+ b²
5(a + b + c)² = a² + b² + c²+ 2ab + 2bc + 2ca
6(a – b – c)² = a²+ b²+ c²– 2ab + 2bc – 2ca
7(a + b)³ = a³+ 3a²b + 3ab²+ b³
8(a + b)³ = a³ + b³ + 3ab(a + b)
9(a – b)³= a³ – 3a²b + 3ab² – b³
10(a – b)³= a³ – b³ – 3ab(a – b)
11a³ – b³ = (a – b)(a²+ ab + b²)
12a³ + b³ = (a + b)(a²– ab + b²)
13(a + b)⁴= a⁴+ 4a³b + 6a²b² + 4ab³ + b²
14(a – b)⁴= a4 – 4a³b + 6a²b² – 4ab³+ b⁴
15a⁴ – b⁴= (a – b)(a + b)(a² + b²)
16a⁵ – b⁵= (a – b)(a⁴ + a³b + a²b² + ab³+ b⁴)

Square of the Sum of Three Terms (a+b+c):

The square of the sum of three terms, a+b+c, expands using the formula for the square of a trinomial:

(a+b+c)2=a2+b2+c2+2ab+2ac+2bc

This formula gives the sum of the squares of each individual term (a^2, b^2, c^2) and twice the product of each pair of terms (2ab, 2ac, 2bc). It applies universally for any values of a, b, and c.

Square of the Difference Between Two Terms (a−b):

The square of the difference between two terms, a−b, expands using the formula for the square of a binomial:

(ab)2=a22ab+b2

Here, the formula provides the square of the first term (a^2), minus twice the product of the two terms (2ab), and the square of the second term (b^2). This expansion holds true for any values of a and b.

Square of the Sum of Two Terms (a+b):

The square of the sum of two terms, a+b, expands using the formula for the square of a binomial:

(a+b)2=a2+2ab+b2

This formula results in the sum of the squares of each individual term (a^2 and b^2) and twice the product of the terms (2ab). It applies universally for any values of a and b.

Example 1: Find the value of 202−15220^2 – 15^2.

Solution: To solve this equation, we use the formula a2−b2=(a+b)(a−b)a^2 – b^2 = (a+b)(a-b).

202−152=(20+15)(20−15)=35×5=175 (Answer)20^2 – 15^2 = (20+15)(20-15) \\ = 35 \times 5 \\ = 175 \text{ (Answer)}

Example 2: Given x−y=2x – y = 2 and x2+y2=20x^2 + y^2 = 20, find the values of xx and yy (where x,y>0x, y > 0).

Solution: From x2+y2=20x^2 + y^2 = 20:

(x−y)2+2xy=2022+2xy=204+2xy=202xy=16xy=8(x-y)^2 + 2xy = 20 \\ 2^2 + 2xy = 20 \\ 4 + 2xy = 20 \\ 2xy = 16 \\ xy = 8

Now, (x+y)2=(x−y)2+4xy=4+32=36(x+y)^2 = (x-y)^2 + 4xy = 4 + 32 = 36:

x+y=±6x+y=6 (since x,y>0) (1)x−y=2 (2)x+y = \pm 6 \\ x+y = 6 \text{ (since \( x, y > 0 \))} \text{ (1)} \\ x-y = 2 \text{ (2)}

Solving equations (1) and (2):

x=4,y=2 (Answer)x = 4, \quad y = 2 \text{ (Answer)}

Example 3: Divide a3+b3+c3−3abca^3 + b^3 + c^3 – 3abc by a+b+ca+b+c and find the quotient.

Solution:

a3+b3+c3−3abc=(a+b+c)(a2+b2+c2−ab−ac−bc)Quotient=(a+b+c)(a2+b2+c2−ab−ac−bc)a+b+c=a2+b2+c2−ab−ac−bcMagnitude of quotient is 2 (Answer)a^3 + b^3 + c^3 – 3abc = (a+b+c)(a^2 + b^2 + c^2 – ab – ac – bc) \\ \text{Quotient} = \frac{(a+b+c)(a^2 + b^2 + c^2 – ab – ac – bc)}{a+b+c} \\ = a^2 + b^2 + c^2 – ab – ac – bc \\ \text{Magnitude of quotient is } 2 \text{ (Answer)}

Example 4: Find the successive products (x+y),(x−y),(x2+y2)(x+y), (x-y), (x^2 + y^2).

Solution:

(x+y)(x−y)(x2+y2)=(x2−y2)(x2+y2)=x4−y4 (Answer)(x+y)(x-y)(x^2 + y^2) = (x^2 – y^2)(x^2 + y^2) \\ = x^4 – y^4 \text{ (Answer)}

These examples are now rewritten for clarity and correctness.

  1. Given x+y=3x+y = 3 and xy=2xy = 2, find the value of (x−y)2(x – y)^2.
  2. If a+b=8a+b = 8 and ab=15ab = 15, determine the values of aa and bb.
  3. If a+b=5a+b = 5 and ab=6ab = 6, find the value of a2−b2a^2 – b^2.
  4. For x=29x = 29 and y=14y = 14, calculate 4×2+9y2+12xy4x^2 + 9y^2 + 12xy.

And here are some algebraic identity formulas in Hindi:

बीजगणित सूत्र:

  • a2−b2=(a−b)(a+b)a^2 – b^2 = (a – b)(a + b)
  • (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  • a2+b2=(a+b)2−2aba^2 + b^2 = (a + b)^2 – 2ab
  • (a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2
  • (a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
  • (a−b−c)2=a2+b2+c2−2ab−2bc+2ca(a – b – c)^2 = a^2 + b^2 + c^2 – 2ab – 2bc + 2ca

बीजगणित सूत्र घन:

  • (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
  • (a+b)3=a3+b3+3ab(a+b)(a + b)^3 = a^3 + b^3 + 3ab(a + b)
  • (a−b)3=a3−3a2b+3ab2−b3(a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3
  • (a−b)3=a3−b3−3ab(a−b)(a – b)^3 = a^3 – b^3 – 3ab(a – b)
  • a3−b3=(a−b)(a2+ab+b2)a^3 – b^3 = (a – b)(a^2 + ab + b^2)
  • a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 – ab + b^2)

और कुछ अतिरिक्त बीजगणित सूत्र:

  • (a+b)4=a4+4a3b+6a2b2+4ab3+b4(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4
  • (a−b)4=a4−4a3b+6a2b2−4ab3+b4(a – b)^4 = a^4 – 4a^3b + 6a^2b^2 – 4ab^3 + b^4
  • a4−b4=(a−b)(a+b)(a2+b2)a^4 – b^4 = (a – b)(a + b)(a^2 + b^2)
  • a5−b5=(a−b)(a4+a3b+a2b2+ab3+b4)a^5 – b^5 = (a – b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)

Frequently Asked Questions

  • What is the algebraic formula for a+b whole cube?

    • The formula for (a+b)3(a+b)^3 is a3+3a2b+3ab2+b3a^3 + 3a^2b + 3ab^2 + b^3.
  • How can I derive the formula for a+b whole cube?

    • The formula can be derived by expanding (a+b)(a+b)(a+b)(a+b)(a+b)(a+b) using the distributive property and combining like terms.
  • What is the simplified form of a+b whole cube?

    • The simplified form of (a+b)3(a+b)^3 is a3+3a2b+3ab2+b3a^3 + 3a^2b + 3ab^2 + b^3.
  • Is there an alternative way to express a+b whole cube?

    • Yes, (a+b)3(a+b)^3 can also be expressed as a3+b3+3ab(a+b)a^3 + b^3 + 3ab(a+b).
  • Can the formula a+b whole cube be used for negative numbers a+b whole cube?

    • Yes, the formula is valid for any real numbers aa and bb, including negative numbers.
  • Where can I find a PDF containing all algebraic identity formul as a+b whole cube ?

    • You can find PDFs of algebraic identity formulas on educational websites, online libraries, and mathematics resource sites.
  • How can I access a chart of all algebraic identities a+b whole cube ?

    • Charts of algebraic identities are often available in textbooks, online educational platforms, and downloadable resources from academic websites.
  • Are there any apps that provide a list of algebraic formulas a+b whole cube?

    • Yes, several educational apps provide comprehensive lists of algebraic formulas, including Khan Academy, Wolfram Alpha, and Photomath.
  • What is the importance of knowing algebraic identities like a+b whole cube?

    • Knowing algebraic identities helps simplify complex expressions, solve algebraic equations, and understand the properties of polynomials.
  • Can I get a list of algebraic identities formulas in Hindi a+b whole cube?

    • Yes, many educational websites and resources offer algebraic identities formulas in Hindi. You can search for them online or check local educational platforms.

conclusion

The formula for (a+b)3(a + b)^3 is a fundamental algebraic identity expressed as (a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. This identity is crucial for simplifying expressions and solving polynomial equations. Understanding and applying such algebraic identities streamline complex calculations and problem-solving in algebra and other mathematical fields.

To support students and educators, comprehensive resources like PDF charts and lists of algebraic identities are invaluable. These resources compile essential formulas, providing quick reference and reinforcing understanding. They encompass a variety of identities beyond (a+b)3(a + b)^3, such as (a−b)3(a – b)^3 and other binomial expansions, contributing to a solid mathematical foundation.

Accessing a well-organized algebraic identities chart facilitates efficient learning and application. It enables students to tackle algebraic problems with confidence, enhancing their mathematical skills. Such resources are readily available online, ensuring that students can easily obtain and utilize them in their studies. By mastering these identities, learners can significantly improve their problem-solving capabilities and overall performance in mathematics.

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