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:
\[
\begin{array}{r|l}
2 & 32400\\\hline
2 & 16200\\\hline
2 & 8100\\\hline
2 & 4050\\\hline
3 & 2025\\\hline
3 & 675\\\hline
3 & 225\\\hline
3 & 75\\\hline
5 & 25\\\hline
5 & 5\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
32400 = 2^4 \times 3^4 \times 5^2
\]
\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 & 25\\\hline
5 & 5\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
32400 = 2^4 \times 3^4 \times 5^2
\]
4. Exponents to numerical values:
Converting the given \(\text{exponents}\) into \(\text{numerical values}\) by multiplying the \(\text{base}\) repeated number of times as the \(\text{power}\).
Example:
Express \(5^{4}\) in numerical value.
Solution:
\(5^{4}\)
\(= 5 \times 5 \times 5 \times 5\)
\(=625\)