In this article, We will study the basics of Algebra rules. We will be using these rules to solve various Algebra problems. Learn Basics of Algebra Rules

Let’s assume a, b and c are set of variables, Real numbers, or algebraic expressions. Here, are some algebra rules:

### The Axioms of Equality Algebra Rules

Reflexive or Identity property states that for every real no. a,
Then a = a
Symmetry property states that for all real no. a and b
If a = b, then b = a.
Transitivity property states that for all real number a,b,c,
If a = b and b = c, then a = c. Transitivity
Substitution property states that if x=y then x can be replaced by y in any equation or expression.

#### Commutative rule of addition (basic of algebra rules)

Changing the order of addends doesn’t change the sum in a given equation. In the below equation a,b are real numbers or algebraic variables.
a+b=b+a

For example:

2+6=6+2 or x+1=1+x

#### Commutative rule of multiplication

The commutative property in the basics of  Algebra rule is defined as changing the order or state of factors that don’t change the product of the product.
a\times b=b\times a

For example:

2\times 6= 6\times 2 or x\times 1= 1\times x

#### Associative rule of addition

Re-arranging of addends doesn’t change the sum in a given equation.
(???? + ????) + ???? = ???? + (???? + ????)

For example:

(2+6)+7=2+(6+7) or (x^2+x)+1=x^2+(x+1)

#### Associative rule of multiplication

In Associative rule of multiplication, we group the factors in such a way that the problem doesn’t changes the product. The grouping of number help us to solve the problem easily and quickly.
(a\times b) \times c= a\times (b\times c)

For example:

(2\times 6) \times 4= 2\times (6\times 4) or (x^2 \times x) \times 1= x^2 \times (x\times 1)

#### Distributive rule of addition over multiplication

a \times (b+c) = a \times b + a \ times c

For example:

2\ times (4+6) = 2 \times 4 + 2 \times 6 or x \ times (x+1) =x \ times x +x \times 1

0 is the identity element in addition to the real numbers. For any real number a,
a+0=0+a=a

#### Multiplicative Identity

1 is the identity element in addition to the real numbers. For any real number a,
a\times 1= 1\times a=a

#### The additive inverse

a+(-a) = -a+a=0
So, the additive inverse of a is -a.

#### The multiplicative reciprocal or inverse rule

When the product of two number is one, the are called reciprocal or inverse property.
a \times \frac{1}{a} =\frac{a}{a} =1 Likewise, the reciprocal of \frac{x}{y} = \frac{y}{x}

#### Algebraic definition of subtraction and division

a-b= a+(-b)
Subtraction in algebra is defined as additive inverse.

a \div b= \frac{a}{b}=a \times \frac{1}{b} where, b \neq0 Division, in algebra is defined as multiplicative reciprocal.

-(-a) =a

#### Rule for Zero

a \times 0= 0 \times a =0

\frac{0}{a} =0 but \frac {a}{0} is not defined where a \neq0

#### Same operator on both side of the Algebra equations

When we add or multiply the same number or variable on both the side of the equation.
If a=b then, a+c = b+c
If a=b then, ac=bc

#### Change of sign on both side of equation

We can change the sign on both sides of the equation
If -a=b then, a= -b

#### Change of sign-on both side of inequality

When we change the sign on both sides of the equation then we must change the direction of the inequality.
If a>b, then -a< -b.

#### Change of the sense in an inequality

if -ay< \frac{1}{b}   then, y> - \frac{1}{ab}

#### Addition of fraction

In case of same denominator, \frac{a}{b} + \frac{c}{b}=\frac{a+c}{b} In case of different denominator, \frac{a}{b} + \frac{c}{d}=\frac{ad+cb}{db}

#### Subtraction of fraction

In case of same denominator, \frac{a}{b} - \frac{c}{b}=\frac{a-c}{b} In case of different denominator, \frac{a}{b} - \frac{c}{d}=\frac{ad-cb}{db}

#### Multiplication of fraction

\frac{a}{c} \times \frac{b}{d}=\frac{ab}{cd}

#### Division of fraction

\frac{a}{c} \div \frac{b}{d}=\frac{a}{c} \times \frac{d}{b}=\frac{ad}{cb}

#### Power and exponents

Let n be the natural number, then a^n= a \times a \times a \times a\dotso \times a

a is the law of exponents

Algebra rules with few examples :
x^1 = x x^0 = 1 x^m x^n= x^{m+n} \frac{x^m}{x^n}= x^{m-n} (x^m)^n= x^{mn} x^{-n}= \frac{1}{x^n} (\frac{x}{y})^n= \frac{x^n}{y^n} x^{\frac{1}{n}} =\sqrt[n]{x} x^{\frac{m}{n}} =\sqrt[n]{x^m}

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