Rational and Radical Checkpoint
A free College Algebra lesson from the “Rational Expressions and Equations” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint reviews exponent rules, radicals, rational expressions, rational equations, coordinate formulas, and conic basics.
What you'll learn
- Simplify radicals and rational expressions
- Solve rational and radical equations
- Use exponent rules
Why it matters: This checkpoint focuses on expressions that can restrict inputs or create extraneous answers. It helps students prove they can simplify carefully while preserving the domain and checking candidate solutions.
Worked example
Problem. Simplify (x^2 - 16)/(x - 4).
- Factor x^2 - 16.
- Cancel x - 4.
- Keep x != 4 as a restriction.
Answer: x + 4, x != 4
Practice problems
1. Simplify sqrt(75).
Show solution
- Radicals: First identify exactly what the question is asking: Simplify sqrt(75).
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- 75 = 25 x 3.
- Check the result by substituting or estimating: the response should match 5sqrt(3) and make sense in the original problem.
Answer: 5sqrt(3)
2. Simplify (x^2 - 36)/(x - 6).
Choices: x + 6, x != 6 · x - 6 · x + 6, x != -6 · x^2 + 6
Show solution
- Rational Expressions: First identify exactly what the question is asking: Simplify (x^2 - 36)/(x - 6).
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Factor difference of squares.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x + 6, x != 6
3. Solve 15/(x + 2) = 5.
Show solution
- Rational Equations: First identify exactly what the question is asking: Solve 15/(x + 2) = 5.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- 15 = 5(x + 2).
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
4. Find the distance between (0, 0) and (5, 12).
Show solution
- Geometry: First identify exactly what the question is asking: Find the distance between (0, 0) and (5, 12).
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use the 5-12-13 triangle.
- Check the result by substituting or estimating: the response should match 13 and make sense in the original problem.
Answer: 13
5. Simplify x^4 * x^3.
Choices: x^7 · x^12 · x · 2x^7
Show solution
- Exponents: First identify exactly what the question is asking: Simplify x^4 * x^3.
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Add exponents with the same base.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x^7
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