What is the simplest form of $\frac{2 \sqrt{3}}{\sqrt{6}}$?
Answer (2)
The simplest form of 6 2 3 is 2 after rationalizing the denominator and simplifying the expression. This involves multiplying the numerator and denominator by 6 and simplifying the resulting expression. Finally, the common factors are canceled out to arrive at the final result.
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Multiply the numerator and denominator by 6 to rationalize the denominator: 6 2 3 = 6 × 6 2 3 × 6 .
Simplify the numerator: 2 3 × 6 = 2 18 = 2 × 3 2 = 6 2 .
Simplify the fraction: 6 6 2 = 2 .
The simplest form of the expression is 2 .
Explanation
Understanding the Problem We are given the expression 6 2 3 and asked to simplify it. We need to simplify this expression and compare it to the given options: 2 , 3 , 3 2 , and 2 5 .
Rationalizing the Denominator To simplify the expression, we can multiply the numerator and the denominator by 6 to rationalize the denominator: 6 2 3 = 6 × 6 2 3 × 6 = 6 2 18 .
Simplifying the Square Root Now, we simplify the square root in the numerator. Since 18 = 9 × 2 , we have 18 = 9 × 2 = 9 × 2 = 3 2 . Substituting this back into the expression, we get: 6 2 18 = 6 2 × 3 2 = 6 6 2 .
Simplifying the Fraction Finally, we simplify the fraction by canceling the common factor of 6 in the numerator and the denominator: 6 6 2 = 2 .
Final Answer Comparing the simplified expression 2 with the given options, we see that it matches the first option. Therefore, the simplest form of 6 2 3 is 2 .
Examples
Simplifying radical expressions is a fundamental skill in algebra and is used in various fields such as physics, engineering, and computer graphics. For example, when calculating distances or areas involving geometric shapes, you often encounter radical expressions that need to be simplified to obtain the most accurate and manageable result. Imagine you are designing a triangular garden bed and need to calculate the length of a side using the Pythagorean theorem, which results in a radical expression. Simplifying this expression allows you to determine the exact amount of fencing material needed, saving time and resources.
