Exponent Rules

Product Rule [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]

Quotient Rule [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

Power Rule [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]

Zero Exponent [latex]{a}^{0}=1[/latex]

Negative Exponent [latex]{a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}[/latex]

Power of a Product [latex]\large{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex]

Power of a Quotient [latex]\large{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}[/latex]

Slope-Intercept Form

Perhaps the most familiar form of a linear equation is slope-intercept form written as [latex]y=mx+b[/latex], where [latex]m=\text{slope}[/latex] and [latex]b=y\text{-intercept}[/latex].

Fractions

When simplifying algebraic expressions, we may sometimes need to add, subtract, simplify, multiply, or divide fractions. It is important to be able to do these operations on the fractions without converting them to decimals.

• To multiply fractions, multiply the numerators and place them over the product of the denominators.
  •  [latex]\dfrac{a}{b}\cdot\dfrac{c}{d} = \dfrac {ac}{bd}[/latex]
• To divide fractions, multiply the first by the reciprocal of the second.
  •  [latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}[/latex]
• To simplify fractions, find common factors in the numerator and denominator that cancel.
•  [latex]\dfrac{24}{32}=\dfrac{2\cdot2\cdot2\cdot3}{2\cdot2\cdot2\cdot2\cdot2}=\dfrac{3}{2\cdot2}=\dfrac{3}{4}[/latex]

To add or subtract fractions, first rewrite each fraction as an equivalent fraction such that each has a common denominator, then add or subtract the numerators and place the result over the common denominator.

•  [latex]\dfrac{a}{b}\pm\dfrac{c}{d} = \dfrac{ad \pm bc}{bd}[/latex]