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Appendix E Properties of Numbers
Associative Laws.
- Addition:
If \(a\text{,}\) \(b\text{,}\) and \(c\) are any numbers, then \((a + b) + c = a + (b + c)\text{.}\)
- Multiplication
If \(a\text{,}\) \(b\text{,}\) and \(c\) are any numbers, then \((a\cdot b)\cdot c = a\cdot (b\cdot c)\text{.}\)
Commutative Laws.
- Addition:
If \(a\) and \(b\) are any numbers, then \(a + b = b + a\text{.}\)
- Multiplication
If \(a\) and \(b\) are any numbers, then \(a\cdot b = b\cdot a\text{.}\)
Distributive Law.
\(a(b + c) = ab + ac\) for any numbers \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\)
Properties of Equality.
- Addition:
If \(a = b\) and \(c\) is any number, then \(a + c = b + c\text{.}\)
- Subtraction:
If \(a = b\) and \(c\) is any number, then \(a - c = b - c\text{.}\)
- Multiplication
If \(a = b\) and \(c\) is any number, then \(a\cdot c = b\cdot c\text{.}\)
- Division
If \(a = b\) and \(c\) is any nonzero number, then \(\frac{a}{c} =\frac{b}{c} \text{.}\)
Fundamental Principle of Fractions.
If \(a\) is any number, and \(b\) and \(c\) are nonzero numbers, then \(\displaystyle{\frac{a\cdot c}{b\cdot c}= \frac{a}{b}}\text{.}\)
Laws of Exponents.
\(\displaystyle a^m\cdot a^n = a^{m+n}\)
\(\displaystyle \left(a^m\right)^n=a^{m+n}\)
\(\displaystyle (ab)^n=a^n b^n\)
\(\displaystyle \displaystyle{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} }\)
Product Rule for Radicals.
If \(a\) and \(b\) are both nonnegative, then \(\sqrt{ab}=\sqrt{a}\sqrt{b} \text{.}\)
Quotient Rule for Radicals.
If \(a\ge 0\) and \(b\gt 0\text{,}\) then \(\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}} \text{.}\)
Zero-Factor Principle.
If \(ab= 0\) then either \(a= 0\) or \(b=0 \text{.}\)
Properties of Absolute Value.
\begin{equation*}
\begin{aligned}[t]
\abs{a + b} \le \abs{a} + \abs{b} \amp\amp \text{Triangle inequality}\\
\abs{a b} = \abs{a} \abs{b} \amp\amp \text{Multiplicative property }
\end{aligned}
\end{equation*}
