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

**states that for all real no. a and b**

*Symmetry property*If a = b, then b = a.

*states that for all real number a,b,c,*

**Transitivity property**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:**

**Associative rule of addition**

Re-arranging of addends doesn’t change the sum in a given equation.

(𝒂 + 𝒃) + 𝒄 = 𝒂 + (𝒃 + 𝒄)

**For example:**

**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:**

**Distributive rule of addition over multiplication**

a \times (b+c) = a \times b + a \ times c
**For example:**

**Additive Identity**

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.

** The inverse of the inverse**

-(-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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