Question

For Individual or Group Work (Exercises 147 – 150)In calculus, it is sometimes desirable to rationalize a numerator. To do this, we multiply the numerator and the denominator by the conjugate of the numerator. For example, (6 - √2)/4 = (6 - √2)/4 × (6 + √2)/(6 + √2) = (36 - 2)/(4(6 + √2)) = 34/(4(6 + √2)) = 17/(2(6 + √2)) = 17/(6 + √2).
2√10 + √7 30

Accepted Answer

Identify the expression to rationalize: the numerator is $$2\sqrt{10} + \sqrt{7}$$ and the denominator is 30. To rationalize the numerator, multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of $$2\sqrt{10} + \sqrt{7} is$$ $$2\sqrt{10} - \sqrt{7}. $$Set up the multiplication: multiply numerator and denominator by $$2\sqrt{10} - \sqrt{7}$$, so the expression becomes $$\frac{(2\sqrt{10} + \sqrt{7})(2\sqrt{10} - \sqrt{7})}{30(2\sqrt{10} - \sqrt{7})}. $$Use the difference of squares formula for the numerator: $$(a + b)(a - b) = a^2$ - b^2$. Here, a = 2$$\sqrt{10}$$ and b$$ = \sqrt{7}$$, so calculate a^2$ - b^2$. Simplify the numerator by squaring each term: a^2$ = (2\sqrt{10})^2$ and b^2$ = (\sqrt{7})^2$, then subtract to get the rationalized numerator. The denominator remains as 30(2$$\sqrt{10} - \sqrt{7})$$.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Rationalizing the Numerator

Conjugates in Algebra

Difference of Squares Formula