Rationalizing the denominator is a simplification technique used when a fraction has a radical, such as a square root, in the denominator. The goal is to rewrite the fraction so the denominator is a rational number. This matters because rationalized forms are often easier to compare, add, subtract, and use in later algebra steps.
It also helps keep final answers in a standard form expected in many math courses.
The main idea is to multiply the fraction by a special form of 1, so the value does not change. If the denominator is a single square root, multiply by that same square root over itself, such as sqrt(3)/sqrt(3). If the denominator is a binomial with a radical, use the conjugate, such as 5 + sqrt(2) for 5 - sqrt(2).
These products use facts like sqrt(a) · sqrt(a) = a and (a - b)(a + b) = a^2 - b^2 to remove radicals from the denominator.
Key Facts
- Rationalizing means rewriting a fraction so there is no radical in the denominator.
- Multiplying by 1 in the form sqrt(b)/sqrt(b) does not change the value of a fraction.
- For a/sqrt(b), rationalize using a/sqrt(b) · sqrt(b)/sqrt(b) = a sqrt(b)/b.
- For 1/(a + sqrt(b)), multiply by the conjugate: 1/(a + sqrt(b)) · (a - sqrt(b))/(a - sqrt(b)).
- Conjugates multiply using (a + sqrt(b))(a - sqrt(b)) = a^2 - b.
- Always simplify after rationalizing, including reducing fractions and simplifying radicals such as sqrt(12) = 2sqrt(3).
Vocabulary
- Rationalize the denominator
- To rewrite a fraction so that its denominator contains no radical expression.
- Radical
- A symbol such as sqrt that represents a root, most often a square root in this topic.
- Denominator
- The bottom part of a fraction that tells what the numerator is being divided by.
- Conjugate
- For a binomial with a radical, the expression formed by changing the sign between the two terms, such as 3 + sqrt(5) and 3 - sqrt(5).
- Simplify
- To rewrite an expression in an equivalent form with reduced fractions, combined like terms, and simplified radicals.
Common Mistakes to Avoid
- Multiplying only the denominator by a radical is wrong because it changes the value of the fraction. You must multiply both the numerator and denominator by the same nonzero expression.
- Using the same binomial instead of the conjugate is wrong for denominators like 2 + sqrt(3). Multiplying by 2 + sqrt(3) usually leaves a radical in the denominator, while multiplying by 2 - sqrt(3) removes it.
- Forgetting to simplify the numerator after rationalizing can leave an answer unfinished. For example, 6sqrt(5)/10 should be reduced to 3sqrt(5)/5.
- Treating sqrt(a + b) as sqrt(a) + sqrt(b) is wrong because square roots do not distribute over addition. For example, sqrt(9 + 16) = 5, but sqrt(9) + sqrt(16) = 7.
Practice Questions
- 1 Rationalize and simplify: 7/sqrt(5).
- 2 Rationalize and simplify: 4/(3 - sqrt(2)).
- 3 Explain why multiplying 1/(6 + sqrt(7)) by (6 - sqrt(7))/(6 - sqrt(7)) removes the radical from the denominator, but multiplying by (6 + sqrt(7))/(6 + sqrt(7)) does not.