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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:
\(5^{4}\)
\(= 5 \times 5 \times 5 \times 5\)
\(=625\)
Laws of Exponents:
1. Quotient rule:
When dividing powers with the same base, subtract the exponents.
\(n^a \div n^b = n^{(a-b)}\), where \(n \neq 0\)
Example:
\(4^4 \div 4^2 = 4^{(4-2)} = 4^2\)
2. Zero exponent rule:
Any non-zero number raised to the power \(0\) is equal to \(1\).
(i.e) \(a^0 = 1\), where \(a \neq 0\).
Example:
\(5^0 = 1\)
3. Negative exponent rule:
A negative exponent means to take the reciprocal of the base and make the exponent positive.
\(n^{-a} = \frac{1}{n^{a}}\), where \(n \neq 0\).
Example:
\(4^{-2} = \frac{1}{4^{2}}\)
4. Product rule:
When multiplying powers with the same base, add the exponents.
\(n^a \times n^b = n^{(a+b)}\), where \(n \neq 0\).
Example:
\(4^2 \times 4^4 = 4^{(2+4)} = 4^6\)
5. Power of Quotient rule:
When both the numerator and the denominator are raised to the same power, the quotient is also raised to that power.
\(\frac{m^{a}}{n^{a}} = {(\frac{m}{n})}^a\), where \(n \neq 0\).
Example:
\(\frac{5^{2}}{6^{2}} = {(\frac{5}{6})}^2\)
6. Power of a power rule:
When raising a power to another power, multiply the exponents.
\((n^{a})^{b} = n^{ab}\)
Example:
\((4^{2})^{3} = 4^{2\times3} = 4^{6} = 4096\)
7. Power of a product rule:
When a product is raised to a power, raise each factor to that power.
\((m \times n)^a\) \(=\) \(m^a \times n^a\)
Example:
\(5^2 \times 6^2\) \(=\) \((5 \times 6)^2\) \(=\) \(30^2\)
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