Exponential Notation:
Exponential notation is nothing but 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 = \({n^3}\)
Prime factorisation using exponents:
Rewriting the given number as a \(\text{product of prime exponents}\) is called \(\text{prime factorization using exponents.}\)
Example:
Express \(324\) using prime factors.
Solution:
\(
\begin{array}{r|l}
2 & 324\\\hline
2 & 162\\\hline
3 & 81\\\hline
3 & 27\\\hline
3 & 9\\\hline
3 & 3\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
324 = 2^2 \times 3^4
\)
\begin{array}{r|l}
2 & 324\\\hline
2 & 162\\\hline
3 & 81\\\hline
3 & 27\\\hline
3 & 9\\\hline
3 & 3\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
324 = 2^2 \times 3^4
\)
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:
\(5^{4}\)
\(= 5 \times 5 \times 5 \times 5\)
\(=625\)
Power Laws:
| Name of the Law | Rule | Example |
| Product of power with same base | \(n^a \times n^b = n^{(a+b)}\) | \(4^2 \times 4^4 = 4^{(2+4)} = 4^6\) |
| Quotient of power with same base | \(n^a \div n^b = n^{(a-b)}\) | \(4^4 \div 4^2 = 4^{(4-2)} = 4^2\) |
| Power of a power | \((n^{a})^{b} = n^{ab}\) | \((4^{2})^{3} = 4^{2\times3} = 4^{6} = 4096\) |
| Power of a product | \((nm)^a = n^a \times m^a\) | \(15^2 = (3 \times 5)^2 = 3^2 \times 5^2\) |
| Power of a quotient | \(\frac{m^a}{n^a} = {(\frac{m}{n})}^a\) | \(\frac{5^{2}}{6^{2}} = {(\frac{5}{6})}^2\) |
| Product of powers with the same exponent | \(m^a \times n^a\) \(=\) \((m \times n)^a\) | \(5^2 \times 6^2\) \(=\) \((5 \times 6)^2\) \(=\) \(30^2\) |
| Quotient of powers with the same exponent | \( \frac{m^a} {n^a}\) \(=\) \((\frac{m}{n})^a\) | \(\frac{3^2} {2^2}\) \(=\) \((\frac{3}{2})^2\) |
| Zero exponent | \(a^0 = 1\), \(a \neq 0\) | \(5^0 = 1\) |
| Negative exponent | \(n^{-a} = \frac{1}{n^{a}}\) | \(4^{-2} = \frac{1}{4^{2}}\) |
What is a combination?
When we have different choices for different parts of something, we can find the total number of possible combinations by multiplying the number of choices.
If there are:
- \(a\) choices for the first item, and
- \(b\) choices for the second item,
then the total number of combinations is: \(a\times b\). If there are more choices: \(a\times b\times c\times\cdots\)
Example:
1. Renu has \(4\) dresses and \(3\) caps.
For each dress, she can choose any of the \(3\) caps. \(4 \times 3 = 12\)
So, there are \(12\) different combinations.
2. A \(6\)-digit lock with \(10\) choices for each digit has \(10\times10\times10\times10\times10=10^5\) possible passwords.
What is a power line?
A power line arranges powers of the same base in order so that we can see how the value changes when the exponent increases or decreases.
Base \(4\):
\(4^{−2},\quad4^{-1},\quad4^0,\quad4^1,\quad4^2,\quad4^3\)
Their values are:
\(\frac{1}{16},\quad\frac{1}{4},\quad{1},\quad4,\quad16,\quad64\)