What is the following quotient?
$\frac{3 \sqrt{8}}{4 \sqrt{6}}$
Answer (2)
Simplify the radicals: Rewrite 8 as 2 2 and 6 as 2 3 .
Substitute and cancel: Substitute the simplified radicals into the expression and cancel the common factor 2 .
Rationalize the denominator: Multiply the numerator and denominator by 3 to rationalize the denominator.
Simplify: Simplify the resulting fraction to obtain the final answer: 2 3 .
Explanation
Understanding the Problem We are given the expression 4 6 3 8 and asked to simplify it and find the equivalent expression from the options.
Listing the Options The options are 5 12 2 − 6 3 , 24 3 6 − 4 3 , 12 3 , 2 3
Solution Strategy We will simplify the expression 4 6 3 8 by simplifying the radicals.
Simplifying Radicals Rewrite 8 as 4 ⋅ 2 = 2 2 and 6 as 2 ⋅ 3 = 2 3 .
Substituting Values Substitute these values into the expression: 4 2 3 3 ( 2 2 ) = 4 2 3 6 2 .
Cancelling Common Factors Cancel out the common factor 2 from the numerator and denominator: 4 3 6 = 2 3 3 .
Rationalizing the Denominator Rationalize the denominator by multiplying both the numerator and denominator by 3 : 2 3 3 3 3 = 2 ( 3 ) 3 3 = 6 3 3 .
Simplifying the Fraction Simplify the fraction: 2 3 .
Finding the Matching Option Compare the simplified expression with the given options and identify the matching one. The simplified expression is 2 3 , which matches the last option.
Examples
Imagine you are a chef and need to scale a recipe. The original recipe calls for 8 ounces of one spice and 6 ounces of another. To maintain the flavor profile, you need to ensure the ratio of these spices remains constant. Simplifying the ratio 6 8 helps you accurately scale the recipe without altering the taste. This type of simplification is crucial in cooking, baking, and even in scientific experiments where maintaining proportions is key to a successful outcome.
The quotient 4 6 3 8 simplifies to 2 3 after simplifying the radicals, canceling common factors, and rationalizing the denominator.
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