1. Exponential growth (Paper Folding Concept):
Folding paper explains the concept behind exponential growth, that is each fold of a paper increases its thickness twice.
This is the power of \(\text{multiplicative growth}\) also called as \(\text{exponential growth}\).
2. Exponential Notations:
Exponential notation is nothing but a repeated multiplication.
Let us consider a number \(3^{4}\) here 3 is called as \(\text{base}\) and 4 is called as \(\text{power}\).
Where, \(3^{4}\) is called as \(\text{exponent}\).
Example:
Square numbers: \({n^2}\); cube numbers = \({n^3}\)
3. Prime factorization using exponents:
Rewriting the given number as a \(\text{product of prime exponents}\) is called \(\text{prime factorization using exponents.}\)
Example:
Express 32400 using prime factors.
Solution:
\[
\begin{array}{r|l}
2 & 32400\\\hline
2 & 16200\\\hline
2 & 8100\\\hline
2 & 4050\\\hline
3 & 2025\\\hline
3 & 675\\\hline
3 & 225\\\hline
3 & 75\\\hline
5 & 15\\\hline
5 & 3\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
32400 = 2^4 \times 3^4 \times 5^2
\]
\begin{array}{r|l}
2 & 32400\\\hline
2 & 16200\\\hline
2 & 8100\\\hline
2 & 4050\\\hline
3 & 2025\\\hline
3 & 675\\\hline
3 & 225\\\hline
3 & 75\\\hline
5 & 15\\\hline
5 & 3\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
32400 = 2^4 \times 3^4 \times 5^2
\]
4. Rules on Multiplying and dividing powers with same base:
Rule for multiplying powers,
\(n^a \times n^b = n^{(a+b)}\)
Rule for dividing powers,
\(n^a \div n^b = n^{(a-b)}\)
Example:
\(4^2 \times 4^4 = 4^{(2+4)} = 4^6\)
\(4^4 \div 4^2 = 4^{(4-2)} = 4^2\)
5. Power of a power:
Rule for calculating a power of a power,
\((n^{a})^{b} = n^{ab}\)
Example:
\((4^{2})^{3} = 4^{2\times3} = 4^{6} = 4096\)
6. When Zero is in power:
Any non zero value is raised to the power 0 it becomes 1.
(i.e) \(a^0 = 1\)
Example:
\(5^0 = 1\)