What is the following quotient?
[tex]\frac{3 \sqrt{8}}{4 \sqrt{6}}[/tex]
A. [tex]\frac{12 \sqrt{2}-6 \sqrt{3}}{5}[/tex]
B. [tex]\frac{3 \sqrt{6}-4 \sqrt{3}}{24}[/tex]
C. [tex]\frac{\sqrt{3}}{12}[/tex]
D. [tex]\frac{\sqrt{3}}{2}[/tex]
Answer (2)
To simplify the quotient 4 6 3 8 , we simplify the square roots and rationalize the denominator. After calculations, we find the final answer is 2 3 , which corresponds to option D.
;
Simplify 8 to 2 2 and 6 to 2 3 .
Substitute these into the expression: 4 2 3 3 ( 2 2 ) .
Cancel 2 and simplify: 4 3 6 = 2 3 3 .
Rationalize the denominator: 2 ( 3 ) 3 3 = 2 3 .
The final answer is 2 3 .
Explanation
Understanding the Problem We are given the expression 4 6 3 8 and asked to simplify it and choose the correct answer from the list of options.
Listing the Options The options are 5 12 2 − 6 3 , 24 3 6 − 4 3 , 12 3 , 2 3
Simplifying Radicals Simplify the expression 4 6 3 8 by simplifying the radicals. Rewrite 8 as 4 ⋅ 2 = 2 2 .
Rewrite 6 as 2 ⋅ 3 = 2 3 .
Substituting Back Substitute these back into the original expression: 4 2 3 3 ( 2 2 ) = 4 2 3 6 2 .
Cancelling Terms Cancel the 2 terms: 4 3 6 = 2 3 3 .
Rationalizing the Denominator Rationalize the denominator by multiplying 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 Answer Compare the simplified expression with the given options and choose the correct one. The simplified expression is 2 3 .
Examples
When simplifying radical expressions, you're essentially making them easier to understand and work with. This is useful in various fields, such as engineering, where you might need to calculate the length of a diagonal in a structure or the distance between two points. Simplifying radicals helps in obtaining more manageable and accurate results in these real-world applications.
