Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Rationalize Denominator
5:30 minutes
Problem 157
Textbook Question
Textbook QuestionRationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 5√x / (2√x + √y)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rationalizing the Denominator
Rationalizing the denominator involves eliminating any radical expressions from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator is a binomial involving a square root, one would multiply by the conjugate of that binomial.
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Conjugates
The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the radical in the denominator. This technique is essential for rationalizing denominators that contain two terms.
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Properties of Exponents and Radicals
Understanding the properties of exponents and radicals is crucial for manipulating expressions involving square roots. Key properties include the fact that √(a*b) = √a * √b and that (√a)^2 = a. These properties allow for simplification of expressions and are particularly useful when rationalizing denominators, as they help in rewriting and simplifying the resulting expressions.
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