Question

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. ﻿3m2+m+n\frac{3m}{2 + \sqrt{m + n}}
2+m+n​3m​

Accepted Answer

Identify the expression to rationalize: $$\frac{3m}{2 + (\sqrt{m} + n)}. $$Notice that the denominator is a sum involving a square root term $$\sqrt{m}$$ and another variable $$n. $$Rewrite the denominator to clarify its structure: $$2 + \sqrt{m} + n. $$Since the denominator is a sum of three terms, consider grouping the radical term with one of the constants to apply rationalization techniques. To rationalize the denominator, multiply both numerator and denominator by the conjugate of the denominator. The conjugate changes the sign of the radical term. Here, the conjugate of $$2 + \sqrt{m} + n$$ can be considered as $$2 + n - \sqrt{m}. $$Multiply numerator and denominator by the conjugate: $$\frac{3m}{2 + \sqrt{m} + n} 	imes \frac{2 + n - \sqrt{m}}{2 + n - \sqrt{m}}. $$This will eliminate the square root in the denominator when you expand the product in the denominator. Expand the denominator using the difference of squares formula: $$(a + b)(a - b) = a^2$ - b^2$, where a = 2 + n$ and b$$ = \sqrt{m}$$. Then simplify the numerator by distributing 3m$ over the conjugate terms.

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. $\(\frac{3m}{2 + \sqrt{m + n}\)}$

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Key Concepts

Rationalizing the Denominator

Conjugates of Binomials

Properties of Square Roots and Nonnegative Variables

Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 3m2+m+n\(\frac{3m}{2 + \sqrt{m + n}\)}2+m+n3m