Rationalize the denominator: 3/(√5 - √2)

Question

Accepted Answer

√5 + √2. The correct answer is √5 + √2. Multiply numerator and denominator by the conjugate (√5 + √2): 3(√5 + √2)/[(√5 - √2)(√5 + √2)] = 3(√5 + √2)/(5 - 2) = 3(√5 + √2)/3 = √5 + √2.

The expression (√5 + √2) without the coefficient 3 in the numerator would only work if the denominator simplified to 3, which is exactly what happens: (√5)² - (√2)² = 5 - 2 = 3, so the 3s cancel. The expression 3(√5 + √2)/3 is correct but not fully simplified; the 3s cancel to give √5 + √2. The expression 3(√5 + √2)/7 incorrectly computes (√5)² - (√2)² as 7 instead of 3.

Rationalizing denominators containing radicals is a standard algebraic skill tested on the CLEP exam. For denominators of the form √a ± √b, multiply by the conjugate √a ∓ √b. The denominator becomes a - b (difference of squares), eliminating the radicals. This technique also applies to complex number division where the conjugate of a + bi is a - bi.

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