**Various Algebraic Formulas & Properties**

Algebra is a branch of mathematics that deals with both numbers and letters. The value of numbers is fixed and the letters or alphabets represent the unknown quantities in the algebra formula. Below given some Various Algebraic Formulas & Properties.

A combination of numbers, letters, factorials, matrices etc. is used to form an algebraic equation or algebraic formula.

**Here is the list of all the important algebraic formulas:**

- a
^{2}– b^{2}= (a – b)(a + b) - (a+b)
^{2}= a^{2}+ 2ab + b^{2} - a
^{2}+ b^{2}= (a – b)^{2}+ 2ab - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2ac + 2bc - (a – b – c)
^{2}= a^{2}+ b^{2}+ c^{2}– 2ab – 2ac + 2bc - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}; (a + b)^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3} - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3} - (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4}) - (a – b)
^{4}= a^{4}– 4a^{3}b + 6a^{2}b^{2}– 4ab^{3}+ b^{4}) - a
^{4}– b^{4}= (a – b)(a + b)(a^{2}+ b^{2}) - a
^{5}– b^{5}= (a – b)(a^{4}+ a^{3}b + a^{2}b^{2}+ ab^{3}+ b^{4})

**प्राकृतिक संख्या (Natural Numbers) –**a^{n}– b^{n}= (a – b)(a^{n-1}+ a^{n-2}+…+ b^{n-2}a + b^{n-1})**सम संख्या (Even) –**(n = 2k), a^{n}+ b^{n}= (a + b)(a^{n-1}– a^{n-2}b +…+ b^{n-2}a – b^{n-1})**विषम संख्या (Odd) –**(n = 2k + 1), a^{n}+ b^{n}= (a + b)(a^{n-1}– a^{n-2}b +…- b^{n-2}a + b^{n-1})- (a + b + c + …)
^{2}= a^{2}+ b^{2}+ c^{2}+ … + 2(ab + ac + bc + …. **घातांक के नियम (Low Of Formula Exponents)**(a

1.^{m})(a^{n}) = a^{m+n}**2.**(ab)^{m}= a^{m}b^{m}**3.**(a^{m})^{n}= a^{mn}

**Few Algebraic Properties**

**Addition’s commutative property**–

**a + b = b + a**

If the order of the elements is modified, the sum of the expression does not change. Expressions or numbers can be used as elements.

**Multiplication’s Commutative Property**–

** a x b = b x a**

The product does not change when the order of the factors is changed. Numbers or phrases can be used as these factors.

**Addition’s Associative Property**–

** (a + b)+ c = a + (b + c)**

The property states that when two or more numbers are brought together to execute basic arithmetic addition, the order of the numbers has no bearing on the outcome.

**Multiplication has an associative property: **

**(a x b) x c = a x (b x c)**

When two or more factors are joined together in fundamental arithmetical multiplication, the order of the elements has no effect on the final result. Also, in this situation, parenthesis is used to organize the items.

**Addition and Multiplication have distributive properties**–

**a × (b + c) = a × b + a × c and (a + b) × c = a × c + b × c**

The distributive property states that multiplying each element by a single term and then adding and subtracting the products is the same as multiplying each element by a single term and then adding and subtracting the products.

**Rule of multiplication over subtraction**–

**p (q-r) = p q – pr**

if p, q, and r are all integers. Similarly, the left and right distributions can be used in the addition rule for multiplication over subtraction.

**Left distributive law if p* (q-r) = (p * q) – (p*r)- and**

**Right distributive law if (p-q) r = (pr) – (q*r)-**

**Algebraic Identities**

Algebraic Identities Math expressions that include numbers, variables (unknown values), and mathematical operations are known as algebraic equations (addition, subtraction, multiplication and division).

Algebraic identities are used in many areas of mathematics, including algebra, geometry, and trigonometry. These are primarily used to find the polynomial factors.