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) ${\mathrm{a}}^{-\mathrm{m}}=\frac{1}{a^m},\left(a^{\mathrm{m}}\ne 0\right)$

(iii) a^m.aⁿ = a^m+n

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

(v) (a^m)ⁿ = a^mn = (aⁿ)^m

(vi) $\sqrt[q]{a^p}={\mathrm{a}}^{\left(\frac{\mathrm{p}}{q}\right)},\mathrm{q}\in \mathrm{N}\text{ and }\mathrm{q}\ge 2$

(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 $\sqrt[n]{x}$ is called surd. The sign '$\sqrt[n]{}$' 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 $\sqrt[2]{x}$ , simply written as $\sqrt{x}$ . 

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

Solution: A conjugate of 3$\sqrt{2}+\sqrt{5}\text{ is }3\sqrt{2}-\sqrt{5}$

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

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

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

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.

1. Addition:

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

Example: 3x + 2x = 5x 

2. Subtraction:

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

 Example: 5x - 2x = 3x 

3. Multiplication:

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

Example: 2x.3y = 6xy 

4. Division:

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

Example: $\frac{6x^2}{2x}$=3 x 

5. 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–1 y + x^2n–2 y² – ... + y²ⁿ), n ∈ N 
• x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz – zx)

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

• 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

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

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 + b² and (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.