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:
\(5^2 \times 6^2\) \(=\) \((5 \times 6)^2\) \(=\) \(30^2\)