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:
\(5^{4}\)
\(= 5 \times 5 \times 5 \times 5\)
\(=625\)
5. Rules on Multiplying powers with same base:
Rule for multiplying powers,
\(n^a \times n^b = n^{(a+b)}\)
Example:
\(4^2 \times 4^4 = 4^{(2+4)} = 4^6\)
6.Power of Quoitent rule
\(\frac{m^{a}}{n^{a}} = {(\frac{m}{n})}^a\)
Example:
\(\frac{5^{2}}{6^{2}} = {(\frac{5}{6})}^2\)
7. Power of a power:
Rule for calculating a power of a power,
\((n^{a})^{b} = n^{ab}\)
Example:
\((4^{2})^{3} = 4^{2\times3} = 4^{6} = 4096\)
8. Power of a product rule:
Rule for calculating power of a product:
\(m^a \times n^a\) \(=\) \((m \times n)^a\)
Example:
\(5^2 \times 6^2\) \(=\) \((5 \times 6)^2\) \(=\) \(30^2\)
9. Rules on dividing powers with same base:
Rule for dividing powers,
\(n^a \div n^b = n^{(a-b)}\)
Example:
\(4^4 \div 4^2 = 4^{(4-2)} = 4^2\)
10. When Zero is in power:
Any non zero value is raised to the power \(0\) it becomes \(1\).
(i.e) \(a^0 = 1\)
Example:
\(5^0 = 1\)
11. Negative Exponents:
Negative exponents can be written using the below rule
\(n^{-a} = \frac{1}{n^{a}}\)
Example:
\(4^{-2} = \frac{1}{4^{2}}\)