Laws of Exponents
Properties of Exponents
The laws of exponents are as follows:
- Product Rule: $a^x \cdot a^y = a^{x+y}$
- Quotient Rule: $\frac{a^x}{a^y} = a^{x-y}$
- Power Rule: $(a^x)^y = a^{xy}$
These rules apply regardless of whether $x$ and $y$ are integers, rational, or real numbers.
Example- $(4^2)^5 = 4^{10}$
- $4^6 \times 4^8 = 4^{14}$
- $\frac{5^4}{5^2}=5^2$
$x^{y^z}$ is not the same as $(x^y)^z$
- The first one is the power being applied on $y$, for example
$$2^{2^3}=2^{2\times 2\times 2}=2^8=256$$
- The second is is the power being applied to the entire number $x^y$ for example
$$(2^3)^2 = 2^6 = 64$$
Negative Exponents
if you have $x^{-a}$ for any $a$
$$x^{-a} = \frac{1}{x^a}$$
Example- $2^{-2} = \frac{1}{2^2}=\frac{1}{4}$
- $\frac{1}{3^{-3}}=3^3=27$
- $5^2\times(5^2)^{-2}=5^2\times5^{-4}=5^{-2}=\frac{1}{5^2}=\frac{1}{25}$
Logarithms: An Introduction
Logarithms are one of the inverse functions to exponentiation, the other being radicals.
Let's say we have $a$, $b$, and $c$ such that $a^b = c$.
