Order of operations
- Simplify: \(\frac{18-4}{2}+3(5-2)^2\)
\[ \frac{18-4}{2}+3(5-2)^2 = \frac{14}{2}+3(3)^2 = 7+3(9) = 7+27 = 34 \]
- Simplify: \(2x+5(3x-4)-\frac{6x}{2}\)
\[ 2x+5(3x-4)-\frac{6x}{2} = 2x+15x-20-3x = 14x-20 \]
Solving equations
- Solve for \(x\): \(5x-7=3x+9\)
\[ 5x-7=3x+9 \Rightarrow 2x=16 \Rightarrow x=8 \]
- Solve for \(x\): \(3(2x-1)=5-(x+4)\)
\[ 3(2x-1)=5-(x+4) \Rightarrow 6x-3=1-x \Rightarrow 7x=4 \Rightarrow x=\frac{4}{7} \]
Rules of exponents
- Simplify: \(\frac{5^7}{5^4}\)
\[ \frac{5^7}{5^4} = 5^{7-4} = 5^3 = 125 \]
- Simplify: \((2x^3y^{-2})^3\)
\[ (2x^3y^{-2})^3 = 2^3x^{9}y^{-6} = \frac{8x^9}{y^6} \]
- Simplify: \(\frac{x^4\cdot x^2}{x^6}\)
\[ \frac{x^4\cdot x^2}{x^6} = \frac{x^{4+2}}{x^6} = \frac{x^6}{x^6} = x^0 = 1 \]
- Simplify: \(\left(\frac{2x}{y}\right)^3\)
\[ \left(\frac{2x}{y}\right)^3 = \frac{2^3x^3}{y^3} = \frac{8x^3}{y^3} \]
- Evaluate: \(27^{2/3}\)
\[ 27^{2/3} = \left(27^{1/3}\right)^2 = 3^2 = 9 \]
FOIL & polynomials
- Expand: \((3x-2)(x+4)\)
\[ (3x-2)(x+4) = 3x^2+12x-2x-8 = 3x^2+10x-8 \]
- What is the degree of the polynomial \(4x^5-3x^2+7x-1\), and how many terms does it have?
The degree is 5 (the largest exponent), and the polynomial has 4 terms.
Unit conversions
- Convert 15 miles to kilometers, given 1 mile = 1.609 km.
\[ 15\text{ mi} \times \frac{1.609\text{ km}}{1\text{ mi}} = 24.135\text{ km} \]
- Convert \(250\ \frac{mg}{L}\) to \(\frac{g}{m^3}\), given 1000 mg = 1 g and 1 L = 0.001 m3.
\[ 250\frac{\text{mg}}{\text{L}} \times \frac{1\text{ g}}{1000\text{ mg}} \times \frac{1\text{ L}}{0.001\text{ m}^3} = 250\frac{\text{g}}{\text{m}^3} \]