Exponents and Powers
When the same number is multiplied by itself repeatedly, the product can be written using exponents. An exponential expression (or a power) consists of a base (\(a\)) and an exponent (\(n\)), written as \(a^n\).
\(a^n = \underbrace{a \times a \times a \times \dots \times a}_{n \text{ times}}\)
Definitions:
Base (\(a\)): The number that is being multiplied(the repeated factor).
Exponent / Power / Index (\(n\)): The number of times the base is multiplied by itself.
Example:
In the expression \(10^{9}\)
\(10\) is a base, \(9\) is the exponent.
It reads as "\(10\) raised to the power of \(9\)".
Prime factorisation using exponents:
Prime Factorization is the process of breaking down a composite number into a product of its prime factors.
Rewriting a number as the product of powers of its prime factors is called expressing the number in prime factorised form.
Steps to find the prime factorisation of a number:
- Divide the given number by the smallest possible prime number that divides it completely without leaving a remainder.
- Write down the quotient directly below the original number.
- Repeat the step with the quotient, dividing it by its smallest prime factor.
- Continue this iterative process until the resulting quotient reduces to \(1\).
Example:
Express \(12800\) using prime factors.
Solution:
\[
\begin{array}{r|l}
2 & 12800\\\hline
2 & 6400\\\hline
2 & 3200\\\hline
2 & 1600\\\hline
2 & 800\\\hline
2 & 400\\\hline
2 & 200\\\hline
2 & 100\\\hline
2 & 50\\\hline
5 & 25\\\hline
5 & 5\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
12800 = 2^9 \times 5^2
\]
\begin{array}{r|l}
2 & 12800\\\hline
2 & 6400\\\hline
2 & 3200\\\hline
2 & 1600\\\hline
2 & 800\\\hline
2 & 400\\\hline
2 & 200\\\hline
2 & 100\\\hline
2 & 50\\\hline
5 & 25\\\hline
5 & 5\\\hline
& 1
\end{array}
\quad\Rightarrow\quad
12800 = 2^9 \times 5^2
\]
Exponents to numerical values:
To evaluate an exponential expression and find its numerical value, multiply the base by itself as many times as indicated by the exponent.
\(a^n = \underbrace{a \times a \times a \times \dots \times a}_{n \text{ factors}}\)
Example:
Express \(5^{4}\) as a numerical value.
Solution:
\(5^{4}\)
\(= 5 \times 5 \times 5 \times 5\)
\(=625\)