Exponents around us:
An exponent tells us how many times a number is multiplied by itself.
Exponents appear in many real-life situations.
Example:
A bacterium doubles itself every hour. After \(1\) hour, it doubles into \(2\) bacteria. After \(2\) hours, the \(2\) bacteria doubles into \(4\). After \(7\) hours, the total number of bacteria is \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7\).
Large quantities and Scientific Notation:
Many quantities in science or facts are too large or too small to write conveniently.
Example:
Distance from Earth to Sun is approximately \(14,96,00,000 \ km\)
Mass of an electron is \(0. 000000000000000000000000000000911 \ kg\).
To prevent writing long strings of zeros, we use scientific notation. A number is written in scientific notation(scientific form or standard form) when it is expressed as:
\(m \times 10^n\)
Where:
1. \(m\) is a number greater than or equal to \(1\), but strictly less than \(10\) (\(1 \le m < 10\)).
2. \(n\) is an integer representing the power of \(10\).
Example:
1. \(3430000 = 3.43 \times 10^{6}\)
2. \(0.0059 = 5.9 \times 10^{-3}\)
Growth patterns:
Many things increase over time. However, not all growth happens in the same way.
There are two important types:
- Linear growth
- Exponential growth
Linear Growth:
Linear growth occurs when a quantity increases (or decreases) by the same fixed amount over equal intervals.
Each equal interval adds (or subtracts) the same amount.
Example: \(2\), \(7\), \(12\), \(17\), ..... (adds \(5\) each time).
Exponential Growth:
Exponential growth occurs when a quantity is multiplied by the same constant factor over equal intervals.
Each equal interval is multiplied by the same factor.
Example: \(4\), \(12\), \(36\), \(108\), ....(multiply by \(3\) each time)