To rationalize the denominator of $\frac{2 \sqrt{10}}{3 \sqrt{11}}$, you should multiply the expression by which fraction?
A. $\frac{\sqrt{10}}{\sqrt{10}}$
B. $\frac{\sqrt{11}}{\sqrt{11}}$
C. $\frac{2-\sqrt{10}}{2-\sqrt{10}}$
D. $\frac{3-\sqrt{11}}{3-\sqrt{11}}$
Answer (2)
Identify the denominator: 3 11 .
Determine the rationalizing factor: 11 .
Multiply the expression by 11 11 to rationalize the denominator.
The fraction to multiply by is 11 11 .
Explanation
Understanding the Problem We are given the expression 3 11 2 10 and asked to find the fraction that will rationalize the denominator. Rationalizing the denominator means eliminating the square root from the denominator.
Identifying the Rationalizing Factor The denominator of the expression is 3 11 . To rationalize it, we need to multiply the denominator by a factor that will eliminate the square root. Multiplying 11 by itself will do the trick, since 11 ⋅ 11 = 11 .
Determining the Correct Fraction To keep the value of the expression unchanged, we must multiply both the numerator and the denominator by the same factor. Therefore, we should multiply the given expression by 11 11 .
Verifying the Result Multiplying the original expression by 11 11 gives us: 3 11 2 10 ⋅ 11 11 = 3 ⋅ 11 2 10 ⋅ 11 = 33 2 110 The denominator is now rationalized.
Final Answer Therefore, the fraction we should multiply the expression by is 11 11 .
Examples
Rationalizing the denominator is a technique used to simplify expressions and make them easier to work with, especially when performing further calculations or comparisons. For example, in physics, you might encounter an expression involving a square root in the denominator when calculating the electric field due to a charge distribution. Rationalizing the denominator can help simplify the expression and make it easier to analyze the behavior of the electric field. Suppose you have an expression like 2 5 . To rationalize the denominator, you multiply both the numerator and denominator by 2 to get 2 5 2 . This form is often preferred because it's easier to approximate the value and perform further calculations.
To rationalize the denominator of 3 11 2 10 , we multiply by the fraction 11 11 . This eliminates the square root in the denominator, resulting in a rational expression. Thus, the answer is option B: 11 11 .
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