3.2 Expressions, Geometry, and Trig
Key Takeaways
- Rational-expression restrictions remain even after factors cancel.
- Radicals and rational exponents are interchangeable forms.
- Logarithms ask what exponent produces a value.
- Geometry and trigonometry questions reward formula recognition plus unit control.
Rational Expressions
For rational expressions, factor first and cancel common factors, not common terms. If an original denominator is zero at a value, that value stays excluded even if the factor later cancels. This is one of the most common advanced ALEKS traps.
Radicals and Logs
| Form | Meaning |
|---|---|
| x^(1/2) | Square root of x |
| x^(m/n) | nth root, then m power |
| log_b(x) | Exponent on b gives x |
| sqrt(x) | Principal nonnegative root |
Logarithms and exponentials are inverse ideas. If log_2(8) = 3, that means 2^3 = 8. For even roots, keep the real-number domain restriction in mind.
Geometry and Trig
Know area formulas, perimeter, Pythagorean theorem, and the basic right-triangle ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Draw the triangle and label sides before choosing a formula. Unit consistency matters as much as the formula itself.
Readiness Check
Before reassessing, solve at least one problem that combines skills: a rational equation with restrictions, a radical equation that needs checking, a right-triangle trig setup, and a geometry problem with unit conversion. Mixed practice reveals whether you understand the topic or only remember isolated formulas.
When simplifying (x - 2)/(x^2 - 4), which value must remain excluded from the domain?
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