## Elementary Algebra Review Sheet

### Basic Rules of Algebra

$a+b=b+a$
• Commutative Property of Multiplication
$ab = ba$
$(a+b)+c=a+(b+c)$
• Associative Property of Multiplication
$(ab)c = a(bc)$
• Left Distributive Property
$a(b+c) = ab + ac$
• Right Distributive Property
$(a+b)c = ac+bc$
$a + 0 = a$
• Multiplicative Identity Property
$a\cdot 1 = 1\cdot a = a$
$a+(-a) = 0$
• Multiplicative Inverse Property
$a \cdot \frac{1}{a} = 1, a\neq 0$

### Properties of Equality

If $a=b$, then $a+c=b+c$
• Multiplication Property of Equality
If $a=b$, then $ac=bc$
If $a+c = b+c$, then $a=b$
• Cancellation Property of Multiplication
If $ab=bc$ and $c\neq 0$, then $a=b$

### Zero-Factor Property

• If $ab=0$, then $a=0$ or $b=0$

### Properties of Negation

• Multiplication by $-1$
$(-1)(a) = -a$ and $(-1)(-a) = a$
• Placement of Minus Signs
$(-a)(b) = -(ab) = (a)(-b)$
• Product of Two Opposites
$(-a)(-b)=ab$

### Operations with Fractions

• $\displaystyle \frac{a}{b}\cdot\frac{c}{d} = \frac{a\cdot c}{b\cdot d}$
• $\displaystyle \frac{a}{b}+\frac{c}{d} = \frac{ad+ bc}{b d}$
• $\displaystyle \frac{a/b}{c/d} = \frac{a}{b}\cdot \frac{d}{c}=\frac{a}{b}\div\frac{c}{d}$
• $\displaystyle \frac{a}{b}-\frac{c}{d} = \frac{ad- bc}{b d}$

### Properties of Exponents

• $a^0 = 1$
• $(ab)^m = a^m \cdot b^m$
• $\dfrac{a^m}{a^n} = a^{m-n}$, $a\neq 0$
• $a^{-n} = \dfrac{1}{a^n}$, $a\neq 0$
• $a^m \cdot a^n = a^{m+n}$
• $\left( a^m \right)^n = a^{m\cdot n}$
• $\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$, $b\neq 0$

### Special Products

• Square of a Binomial
$(u+v)^2 = u^2+2uv+v^2$
$(u-v)^2 = u^2-2uv+v^2$
• Difference of Two Squares
$u^2-v^2 = (u+v)(u-v)$
• Difference of Two Cubes
$u^3-v^3 = (u-v)(u^2+uv+v^2)$
• Sum of Two Cubes
$u^3+v^3 = (u+v)(u^2-uv+v^2)$

• Solutions of $ax^2+bx+c=0$
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

## Common Formulas

### Temperature

• $F = \frac{9}{5}C +32$
$F =$ degrees Farenheit
$C =$ degrees Celsius

### Simple Interest

• $I = P\cdot r\cdot t$
$I =$ interest
$P =$ principal
$r =$ annual interest rate
$t =$ time in years

### Distance

• $d = r\cdot t$
$d =$ distance traveled
$r =$ rate of travel
$t =$ time traveled

### Compound Interest

• $A = P\left( 1+\frac{r}{n} \right)^{nt}$
$A =$ distance traveled
$P =$ rate of travel
$r =$ time traveled
$n =$ rate of travel
$rt=$ time traveled