Question

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

Accepted Answer

Identify the problem: The goal is to rationalize the denominator of the given expression $$(7 + 4\sqrt{2}) / (2 - 5\sqrt{2}). $$This means eliminating the square root from the denominator by multiplying both numerator and denominator by the conjugate of the denominator. Determine the conjugate of the denominator: The conjugate of $$2 - 5\sqrt{2} is$$ $$2 + 5\sqrt{2}. $$Multiplying by the conjugate will use the difference of squares formula $$(a - b)(a + b) = a^2 - b^2 to $$eliminate the square root. Multiply both numerator and denominator by the conjugate: Multiply $$(7 + 4\sqrt{2}) by$$ $$(2 + 5\sqrt{2}) in $$the numerator, and $$(2 - 5\sqrt{2}) by$$ $$(2 + 5\sqrt{2}) in $$the denominator. This ensures the value of the expression remains unchanged. Simplify the denominator: Use the difference of squares formula $$(a - b)(a + b) = a^2 - b^2 to $$simplify $$(2 - 5\sqrt{2})(2 + 5\sqrt{2}). $$This will result in $$2^2 - (5\sqrt{2})^2. $$Simplify the numerator: Expand $$(7 + 4\sqrt{2})(2 + 5\sqrt{2})$$ using the distributive property (FOIL method). Combine like terms, ensuring to simplify any terms involving $$\sqrt{2}. $$After simplifying both numerator and denominator, the expression will be rationalized.

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

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Key Concepts

Rationalizing the Denominator

Conjugates

Simplifying Radicals