What are variables and constants in algebra?

Variables are symbols used to represent unknown quantities, while constants are fixed values that do not change.

What are algebraic expressions?

Algebraic expressions are combinations of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation.

What are algebraic equations?

Algebraic equations are statements that show equality between two algebraic expressions. They typically involve solving for the value(s) of variables that make the equation true.

How do you solve linear equations in algebra?

Linear equations are equations of the form ax + b = 0, where a and b are constants and x are variables. To solve such equations, isolate the variable x by applying inverse operations (e.g., adding or subtracting terms, multiplying or dividing by constants).

What are inequalities in algebra?

Inequalities are mathematical statements that compare two expressions using symbols such as > (greater than), < (less than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving inequalities involves finding the range of values that satisfy the given conditions.

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Fundamentals of Algebra

Q: What is algebra?

Algebra is a mathematical discipline that involves the manipulation and study of symbols, often representing numbers and variables, to solve equations and describe mathematical relationships.

Q: What are the basic algebraic operations?

The basic algebraic operations are addition, subtraction, multiplication, division, and exponentiation (raising to a power).

Fundamentals of algebra include basic concepts such as variables, constants, expressions, equations, inequalities, and operations like addition, subtraction, multiplication, and division. Algebra deals with solving equations, manipulating expressions, and understanding relationships between variables. It is a foundational part of Mathematics used extensively in various fields such as Engineering, Physics, Economics, and Computer Science.

1.0Indices

Indices, also known as exponents or powers, are fundamental mathematical concepts used to represent repeated multiplication. They are denoted by superscript numbers attached to a base number, indicating how many times the base should be multiplied by itself. Understanding indices is crucial in algebra and higher mathematics for simplifying expressions, solving equations, and working with various mathematical operations efficiently.

Indices Definition

If 'a' is any non-zero real or imaginary number and 'm' is a positive integer, then a^m= a. a. a …. a (m times). Here, a is referred to as the base, while m is the index, power, or exponent.

Law of Indices

(i) a⁰ = 1,

(ii) $a^{- m} = \frac{1}{a ^{m}}, (a^{m} \neq = 0)$

(iii) a^m.aⁿ = a^m+n

(iv) $\frac{a ^{m}}{a ^{n}} = a^{m - n}, a^{n} \neq = 0$

(v) (a^m)ⁿ = a^mn = (aⁿ)^m

(vi) $q a^{p} = a^{(\frac{p}{q})}, q \in N and q \geq 2$

(vii) If x = y, then a^x = a^y, but the converse may not be true.

Example: [(25)^1/2]³ =

(A) 25 (B) 5 (C) 125 (D) 625

Ans. (C)

Solution:

[(5²)^1/2]³ ⇒ (5)³ ⇒ 125

2.0Surds

If ‘x’ is a rational number, which is not the n^th power (n ∈ N \{1}) of any rational number, then the number x^1/n usually denoted by $n x$ is called surd. The sign ' $n$ ' is called the radical sign. The number in the angular part of the sign, i.e., 'n' is called order of the surd. In case of n = 2 the expression $2 x$ , simply written as $x$ .

Example: Rationalize the denominator of $\frac{1}{3 2 + 5}$ .

Solution: A conjugate of 3 $2 + 5 is 32 - 5$

Therefore, multiplying the conjugate in the numerator and denominator of the given fraction.

$\frac{3 2 - 5}{( 3 2 + 5 ) ( 3 2 - 5 )}$

$= \frac{3 2 - 5}{( 3 2 ) ^{2} - ( 5 ) ^{2}} = \frac{3 2 - 5}{18 - 5} = \frac{3 2 - 5}{13}$

3.0Introduction to Algebra

Algebra is a mathematical discipline that involves the manipulation and study of symbols, often representing numbers and variables, to solve equations and describe mathematical relationships. It is a fundamental part of mathematics and is widely used in various fields such as science, engineering, economics, and computer science.

4.0Five Basic Rules of Algebra

Algebraic operations are fundamental operations in algebra, Mathematical processes used to manipulate algebraic expressions and solve equations. These operations include five basic rules of algebra - addition, subtraction, multiplication, division, and exponentiation. Understanding how to perform these operations is crucial in algebra and lays the foundation for more complex Mathematical concepts.

Addition:

In algebra, addition involves combining like terms by adding their coefficients.

Example: 3x + 2x = 5x

Subtraction:

Subtraction in algebra is similar to addition but involves subtracting coefficients.

Example: 5x - 2x = 3x

Multiplication:

Multiplication in algebra is represented by the multiplication symbol (×) or by placing terms next to each other.

Example: 2x.3y = 6xy

Division:

Division in algebra is represented by the division symbol (/) or by using fractions.

Example: $\frac{6 x ^{2}}{2 x}$ =3 x

Exponentiation:

Exponentiation involves raising a base to an exponent, represented as xⁿ where 'x' is the base and 'n' is the exponent.

Example: x³ = x ⋅ x ⋅ x

These operations are used to simplify algebraic expressions, solve equations, factor polynomials, and manipulate functions. It's important to follow the rules of algebraic operations, such as the order of operations (PEMDAS/BODMAS) and the properties of exponents, to ensure accurate results.

Additionally, algebraic operations are not limited to simple expressions but are extended to complex numbers, matrices, vectors, and other mathematical structures in advanced algebra and related fields. Mastering algebraic operations is a key step in becoming proficient in algebra and related Mathematical disciplines.

5.0Algebraic Formulas

Algebraic formulas are mathematical expressions that represent relationships and rules in algebra. These formulas are essential tools for solving equations, simplifying expressions, and understanding various mathematical concepts. Here are some common algebraic formulas-

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a – b) (a + b)
a³ ± b³ = (a ± b) (a² _∓ ab + b²)
a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)
xy + ay + bx + ab = (x + a)(y + b)
x² + 2xy + y² = (x + y)²
x² + y² + z² + 2xy + 2yz + 2zx = (x + y + z)²
x⁴ + x² + 1 = (x² + 1)² – x² = (x² + x + 1) (x² – x + 1)
xⁿ – yⁿ = (x – y)(x^n–1 + x^n–2 y + x^n–3y²+ … + y^n–1), n ∈ N
x²ⁿ⁺¹ + y²ⁿ⁺¹ = (x + y)(x²ⁿ – x^2n–1y + x^2n–2y² – ... + y²ⁿ), n ∈ N
x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz – zx)

= $\frac{1}{2} (x + y + z) {(x - y)^{2} + (y - z)^{2} + (z - x)^{2}}$

x³ + y³ + z³ + 3(x + y)(y + z)(z + x) = (x + y + z)³

6.0Basic Algebra Examples with Solutions

Example 1: Solve a⁶ – b⁶

Solution: a⁶ – b⁶ = (a²)³ – (b²)³

= (a² – b²) (a⁴ + a² b² + b⁴)

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

Example 2: Solve (2x + 3)²

Solution: (2x + 3)² = (2x)² + 2(2x)(3) + (3)² = 4x² + 12x + 9.

Example 3: Solve x³ – 27y³

Solution: Using the identity: x³ – y³ = (x – y) (x² + xy + y²).

We rewrite the expression as: x³ – (3y)³

⇒ (x – 3y) (x² + (x) (3y) + (3y)²)

⇒ (x – 3y) (x² + 3xy + 9y²)

Example 4: 8x³ + y³ + 27z³ – 18xyz

Solution: 8x³ + y³ + 27z³ – 18xyz = (2x)³ + (y)³ + (3z)³ – 3(2x) (y)(3z)

= (2x + y + 3z) (4x² + y² + 9z² – 2xy – 6xz – 3yz)

Example 5: Factorize (a – b)³ + (b – c)³ + (c – a)³

Solution:Let x = a – b, y = b – c, z = c – a

⇒x + y + z = 0

⇒ x³ + y³ + z³ = 3xyz

⇒ (a – b)³ + (b – c)³ + (c – a)³ = 3(a – b) (b – c) (c – a)

7.0Fundamentals of Algebra Practice Problems

Solve (2³) – 2
Solve $\frac{4 ^{5} \cdot 4 ^{- 2}}{4 ^{3}}$
Find the values of 'x' that satisfy the equation 2^{x – 1} + 2^{x – 2} = 12.
$\frac{98}{2}$ + y = 15. Find the value of y.
Find the value of x. $18 + 32 = x$
Solve the equation for ‘x’ $3 x + 2 = x + 1$
Simplify $\frac{5 + 3}{5 - 3}$
Expand and simplify : (2y)³ – (5)³
Simplify $\frac{x ^{3} + 3 x + 2}{x + 2}$

8.0Solved Questions for Fundamentals of Algebra

Q1. What are algebraic identities?

Ans: Algebraic identities are mathematical equations that are always true, regardless of the values of the variables involved. Examples include (a + b)² = a² + 2ab + b²and (a – b)² = a² – 2ab + b².

Q2. What is factorization in algebra?

Ans: Factorization involves breaking down an algebraic expression into a product of its constituent factors. For example, the quadratic expression x² – 4 can be expressed as (x + 2)(x - 2).

Q3. What are quadratic equations?

Ans: Quadratic equations are algebraic equations written in the format ax² + bx + c = 0, where a, b, and c are constants/ fixed numeric values.

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Fundamentals of Algebra

Fundamentals of algebra include basic concepts such as variables, constants, expressions, equations, inequalities, and operations like addition, subtraction, multiplication, and division. Algebra deals with solving equations, manipulating expressions, and understanding relationships between variables. It is a foundational part of Mathematics used extensively in various fields such as Engineering, Physics, Economics, and Computer Science.

1.0Indices

Indices, also known as exponents or powers, are fundamental mathematical concepts used to represent repeated multiplication. They are denoted by superscript numbers attached to a base number, indicating how many times the base should be multiplied by itself. Understanding indices is crucial in algebra and higher mathematics for simplifying expressions, solving equations, and working with various mathematical operations efficiently.

Indices Definition

If 'a' is any non-zero real or imaginary number and 'm' is a positive integer, then a^m= a. a. a …. a (m times). Here, a is referred to as the base, while m is the index, power, or exponent.

Law of Indices

(i) a⁰ = 1,

(ii) $a^{- m} = \frac{1}{a ^{m}}, (a^{m} \neq = 0)$

(iii) a^m.aⁿ = a^m+n

(iv) $\frac{a ^{m}}{a ^{n}} = a^{m - n}, a^{n} \neq = 0$

(v) (a^m)ⁿ = a^mn = (aⁿ)^m

(vi) $q a^{p} = a^{(\frac{p}{q})}, q \in N and q \geq 2$

(vii) If x = y, then a^x = a^y, but the converse may not be true.

Example: [(25)^1/2]³ =

(A) 25 (B) 5 (C) 125 (D) 625

Ans. (C)

Solution:

[(5²)^1/2]³ ⇒ (5)³ ⇒ 125

2.0Surds

If ‘x’ is a rational number, which is not the n^th power (n ∈ N \{1}) of any rational number, then the number x^1/n usually denoted by $n x$ is called surd. The sign ' $n$ ' is called the radical sign. The number in the angular part of the sign, i.e., 'n' is called order of the surd. In case of n = 2 the expression $2 x$ , simply written as $x$ .

Example: Rationalize the denominator of $\frac{1}{3 2 + 5}$ .

Solution: A conjugate of 3 $2 + 5 is 32 - 5$

Therefore, multiplying the conjugate in the numerator and denominator of the given fraction.

$\frac{3 2 - 5}{( 3 2 + 5 ) ( 3 2 - 5 )}$

$= \frac{3 2 - 5}{( 3 2 ) ^{2} - ( 5 ) ^{2}} = \frac{3 2 - 5}{18 - 5} = \frac{3 2 - 5}{13}$

3.0Introduction to Algebra

Algebra is a mathematical discipline that involves the manipulation and study of symbols, often representing numbers and variables, to solve equations and describe mathematical relationships. It is a fundamental part of mathematics and is widely used in various fields such as science, engineering, economics, and computer science.

4.0Five Basic Rules of Algebra

Algebraic operations are fundamental operations in algebra, Mathematical processes used to manipulate algebraic expressions and solve equations. These operations include five basic rules of algebra - addition, subtraction, multiplication, division, and exponentiation. Understanding how to perform these operations is crucial in algebra and lays the foundation for more complex Mathematical concepts.

Addition:

In algebra, addition involves combining like terms by adding their coefficients.

Example: 3x + 2x = 5x

Subtraction:

Subtraction in algebra is similar to addition but involves subtracting coefficients.

Example: 5x - 2x = 3x

Multiplication:

Multiplication in algebra is represented by the multiplication symbol (×) or by placing terms next to each other.

Example: 2x.3y = 6xy

Division:

Division in algebra is represented by the division symbol (/) or by using fractions.

Example: $\frac{6 x ^{2}}{2 x}$ =3 x

Exponentiation:

Exponentiation involves raising a base to an exponent, represented as xⁿ where 'x' is the base and 'n' is the exponent.

Example: x³ = x ⋅ x ⋅ x

These operations are used to simplify algebraic expressions, solve equations, factor polynomials, and manipulate functions. It's important to follow the rules of algebraic operations, such as the order of operations (PEMDAS/BODMAS) and the properties of exponents, to ensure accurate results.

Additionally, algebraic operations are not limited to simple expressions but are extended to complex numbers, matrices, vectors, and other mathematical structures in advanced algebra and related fields. Mastering algebraic operations is a key step in becoming proficient in algebra and related Mathematical disciplines.

5.0Algebraic Formulas

Algebraic formulas are mathematical expressions that represent relationships and rules in algebra. These formulas are essential tools for solving equations, simplifying expressions, and understanding various mathematical concepts. Here are some common algebraic formulas-

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
a² – b² = (a – b) (a + b)
a³ ± b³ = (a ± b) (a² _∓ ab + b²)
a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)
xy + ay + bx + ab = (x + a)(y + b)
x² + 2xy + y² = (x + y)²
x² + y² + z² + 2xy + 2yz + 2zx = (x + y + z)²
x⁴ + x² + 1 = (x² + 1)² – x² = (x² + x + 1) (x² – x + 1)
xⁿ – yⁿ = (x – y)(x^n–1 + x^n–2 y + x^n–3y²+ … + y^n–1), n ∈ N
x²ⁿ⁺¹ + y²ⁿ⁺¹ = (x + y)(x²ⁿ – x^2n–1y + x^2n–2y² – ... + y²ⁿ), n ∈ N
x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz – zx)

= $\frac{1}{2} (x + y + z) {(x - y)^{2} + (y - z)^{2} + (z - x)^{2}}$

x³ + y³ + z³ + 3(x + y)(y + z)(z + x) = (x + y + z)³

6.0Basic Algebra Examples with Solutions

Example 1: Solve a⁶ – b⁶

Solution: a⁶ – b⁶ = (a²)³ – (b²)³

= (a² – b²) (a⁴ + a² b² + b⁴)

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

Example 2: Solve (2x + 3)²

Solution: (2x + 3)² = (2x)² + 2(2x)(3) + (3)² = 4x² + 12x + 9.

Example 3: Solve x³ – 27y³

Solution: Using the identity: x³ – y³ = (x – y) (x² + xy + y²).

We rewrite the expression as: x³ – (3y)³

⇒ (x – 3y) (x² + (x) (3y) + (3y)²)

⇒ (x – 3y) (x² + 3xy + 9y²)

Example 4: 8x³ + y³ + 27z³ – 18xyz

Solution: 8x³ + y³ + 27z³ – 18xyz = (2x)³ + (y)³ + (3z)³ – 3(2x) (y)(3z)

= (2x + y + 3z) (4x² + y² + 9z² – 2xy – 6xz – 3yz)

Example 5: Factorize (a – b)³ + (b – c)³ + (c – a)³

Solution:Let x = a – b, y = b – c, z = c – a

⇒x + y + z = 0

⇒ x³ + y³ + z³ = 3xyz

⇒ (a – b)³ + (b – c)³ + (c – a)³ = 3(a – b) (b – c) (c – a)

7.0Fundamentals of Algebra Practice Problems

Solve (2³) – 2
Solve $\frac{4 ^{5} \cdot 4 ^{- 2}}{4 ^{3}}$
Find the values of 'x' that satisfy the equation 2^{x – 1} + 2^{x – 2} = 12.
$\frac{98}{2}$ + y = 15. Find the value of y.
Find the value of x. $18 + 32 = x$
Solve the equation for ‘x’ $3 x + 2 = x + 1$
Simplify $\frac{5 + 3}{5 - 3}$
Expand and simplify : (2y)³ – (5)³
Simplify $\frac{x ^{3} + 3 x + 2}{x + 2}$

8.0Solved Questions for Fundamentals of Algebra

Q1. What are algebraic identities?

Ans: Algebraic identities are mathematical equations that are always true, regardless of the values of the variables involved. Examples include (a + b)² = a² + 2ab + b²and (a – b)² = a² – 2ab + b².

Q2. What is factorization in algebra?

Ans: Factorization involves breaking down an algebraic expression into a product of its constituent factors. For example, the quadratic expression x² – 4 can be expressed as (x + 2)(x - 2).

Q3. What are quadratic equations?

Ans: Quadratic equations are algebraic equations written in the format ax² + bx + c = 0, where a, b, and c are constants/ fixed numeric values.

Frequently Asked Questions

What is algebra?

What are variables and constants in algebra?

What are algebraic expressions?

What are algebraic equations?

What are the basic algebraic operations?

How do you solve linear equations in algebra?

What are inequalities in algebra?

Join ALLEN!

Fundamentals of Algebra

1.0Indices

Indices Definition

Law of Indices

2.0Surds

3.0Introduction to Algebra

4.0Five Basic Rules of Algebra

5.0Algebraic Formulas

6.0Basic Algebra Examples with Solutions

7.0Fundamentals of Algebra Practice Problems

8.0Solved Questions for Fundamentals of Algebra

Table of Contents

Frequently Asked Questions

What is algebra?

What are variables and constants in algebra?

What are algebraic expressions?

What are algebraic equations?

What are the basic algebraic operations?

How do you solve linear equations in algebra?

What are inequalities in algebra?

Join ALLEN!

Fundamentals of Algebra

1.0Indices

Indices Definition

Law of Indices

2.0Surds

3.0Introduction to Algebra

4.0Five Basic Rules of Algebra

5.0Algebraic Formulas

6.0Basic Algebra Examples with Solutions

7.0Fundamentals of Algebra Practice Problems

8.0Solved Questions for Fundamentals of Algebra

Table of Contents