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INTRODUCTION OF EXPONENTS & INDICES

__EXPONENTS & INDICES -__

**Exponents and indices are mathematical notations used to represent repeated multiplication of a number by itself or the power to which a number is raised. They are commonly used in algebra and other mathematical disciplines.**

**In the notation "a^b," the letter 'a' is called the base, and 'b' is called the exponent or index. The exponent tells us how many times the base should be multiplied by itself. Here are some key concepts related to exponents and indices:**

**Multiplication with exponents: a^b represents 'a' multiplied by itself 'b' times. For example:**

**2^3 = 2 × 2 × 2 = 8****5^2 = 5 × 5 = 25**

**Negative exponents: When an exponent is negative, it represents the reciprocal of the positive exponent. For example:**

**2^(-3) = 1 / (2^3) = 1 / 8****3^(-2) = 1 / (3^2) = 1 / 9**

**Zero exponent: Any nonzero number raised to the power of zero is equal to 1:**

**a^0 = 1 (where 'a' is non-zero)**

**Exponent rules: There are several rules for working with exponents, which are essential for simplifying expressions and solving equations. Some of the basic rules include:**

**a^m × a^n = a^(m + n) (When multiplying bases with the same value, add the exponents.)****a^m ÷ a^n = a^(m - n) (When dividing bases with the same value, subtract the exponents.)****(a^m)^n = a^(m × n) (When raising an exponent to another exponent, multiply the exponents.)**

**Fractional exponents: An exponent expressed as a fraction represents taking the nth root of the base. For example:**

**4^(1/2) = √4 = 2 (square root of 4)****8^(1/3) = ∛8 = 2 (cube root of 8)**

**Exponents are used in various mathematical contexts, such as algebraic expressions, scientific notation, and calculus. They provide a powerful and compact way to represent repeated operations and large or small numbers. Understanding exponents is fundamental for solving many mathematical problems.**

**Another way of understanding -**

**Exponents and indices are mathematical concepts used to represent repeated multiplication. They are a fundamental part of algebra and play a crucial role in various mathematical calculations.**

**An exponent (also known as a power or index) is a small number written above and to the right of a base number, indicating how many times the base is multiplied by itself. The exponent tells us how many times the base is to be used as a factor in a multiplication operation.**

**The general form of an exponentiation expression is: aⁿ**

**Here:**

**Here, a is the base, the number being multiplied repeatedly.****And, n is the exponent, the number of times the base is multiplied by itself.**

**For Example: 2³= 2 × 2 × 2 = 8**

**In this case, 2 is the base and 3 is the exponent.**

**Key Properties of Exponents:-**

**1.) Multiplication of exponents with the same base: When you multiply two exponential expressions with the same base, you can add their exponents. aᵐ X aⁿ = aᵐ⁺ⁿ or **

__Example.1)__ 2³ X 2⁵ = 2³⁺⁵ = 2⁸

** ; or, 2⁸ = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 256**

**2.) Division of exponents with the same base: When you divide two exponential expressions with the same base, you can subtract their exponents.**

** aᵐ**

** -------- = aᵐ⁻ⁿ**

** aⁿ **

** 3⁸**

__Example.1)__ ------- = 3⁸⁻⁵ = 3³

** 3⁵**

**3.) Exponentiation of a power: When you raise a power to another exponent, you can multiply the exponents.**

** (aᵐ)ⁿ = aᵐ ˣ ⁿ**

__Example.1__) (2³)² = 2³ X 2² = 2³ˣ² = 2⁶

** = 2 X 2 X 2 X 2 X 2 = 32**

**4.) Zero exponent: Any non-zero number raised to the power of zero is equal to 1. **

** a⁰ = 1, (Where, a≠ 0)**

**5) Negative exponent: To represent a negative exponent, take the reciprocal of the base and make the exponent positive.**

** 1**

** a⁻ⁿ = -------**

** aⁿ**

__Example.1)__

** 1 1 1**

** 2⁻⁶ =
-------- = ---------------------- = -------**

** 2⁶ 2 X 2 X 2 X 2 X 2 X 2 64**

**These properties are essential for simplifying and solving equations involving exponents and for understanding more advanced mathematical concepts. Exponents are used in various fields, such as physics, engineering, computer science, and finance, to model various phenomena and make calculations more convenient.**