Quick Revision Notes: Exponents and Powers (घात और घातांक)

Class: 8th | Subject: Mathematics | Board: MP Board (NCERT Based)

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1. Important Definitions (महत्वपूर्ण परिभाषाएं)

• Exponent (घातांक): Tells how many times a number is multiplied by itself.
• Base (आधार): The number being multiplied.
• Notation: In a^n,
  • a is the Base (आधार)
  • n is the Exponent/Power (घात/घातांक)
  • Read as: "a raised to the power of n"

Example: In 5^3, Base = 5, Exponent = 3.
Value: 5^3 = 5 × 5 × 5 = 125

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2. Powers with Negative Exponents (ऋणात्मक घातांक)

For any non-zero integer a and positive integer m:

• a^(-m) = 1 / (a^m)
• 1 / (a^(-m)) = a^m
• (a/b)^(-m) = (b/a)^m

Example:
2^(-3) = 1 / (2^3) = 1 / 8
(2/3)^(-2) = (3/2)^2 = 9/4

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3. Laws of Exponents (घातांक के नियम)

Where a and b are non-zero integers, and m and n are rational numbers/integers:

S.No.	Law / Formula	Rule Name	Example
1	a^m × a^n = a^(m + n)	Multiplying powers with same base	2^3 × 2^4 = 2^(3+4) = 2^7
2	a^m ÷ a^n = a^(m - n)	Dividing powers with same base	5^6 ÷ 5^2 = 5^(6-2) = 5^4
3	(a^m)^n = a^(m × n)	Power of a power	(3^2)^4 = 3^(2×4) = 3^8
4	a^m × b^m = (a × b)^m	Multiplying powers with same exponent	2^3 × 5^3 = (2 × 5)^3 = 10^3
5	a^m ÷ b^m = (a / b)^m	Dividing powers with same exponent	8^2 ÷ 4^2 = (8 / 4)^2 = 2^2
6	a^0 = 1	Zero exponent (Any number to power 0 is 1)	7^0 = 1, (-100)^0 = 1
7	(-1)^even = 1	Negative one raised to an even power	(-1)^4 = 1
8	(-1)^odd = -1	Negative one raised to an odd power	(-1)^5 = -1

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4. Standard Form vs. Usual Form (मानक रूप एवं सामान्य रूप)

A. Standard Form (Scientific Notation)

A number is written in standard form if it is expressed as:

k × 10^n

• Where 1 <= k < 10 (k is a decimal number from 1 to 9.99...)
• n is an integer (can be positive or negative)

B. Rules for Conversion:

1. Very Large Numbers (Positive Power of 10):

   • Move the decimal point to the LEFT until you get a number between 1 and 10.
   • Count the number of places moved; this is your positive exponent n.
   • Example: 150,000,000 = 1.5 × 10^8

2. Very Small Numbers (Negative Power of 10):

   • Move the decimal point to the RIGHT until you get a number between 1 and 10.
   • Count the number of places moved; this is your negative exponent -n.
   • Example: 0.000007 = 7.0 × 10^(-6)

C. Standard Form to Usual Form:

• If power of 10 is positive, move the decimal to the RIGHT.
  • Example: 3.4 × 10^5 = 340,000
• If power of 10 is negative, move the decimal to the LEFT.
  • Example: 4.5 × 10^(-4) = 0.00045

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5. Quick Revision Tips for Board Exams

1. Sign Mistake: (-2)^4 is NOT the same as -2^4.
   • (-2)^4 = (-2) × (-2) × (-2) × (-2) = +16
   • -2^4 = -(2 × 2 × 2 × 2) = -16
2. Reciprocal Property: To change a negative power to a positive power, simply invert the base: (a/b)^(-m) = (b/a)^m.
3. Always reduce fractions and evaluate the power completely in final answers unless instructed otherwise.