What is the following quotient?
[tex]\frac{3 \sqrt{8}}{4 \sqrt{6}}[/tex]
Answer (2)
The quotient of 4 6 3 8 simplifies to 2 3 . This is achieved by simplifying the numerator and denominator separately and rationalizing the result. The final answer is 2 3 .
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Simplify Expression 1: 4 6 3 8 = 2 3
Simplify Expression 2: 5 3 12 2 − 6 3
Divide Expression 1 by Expression 2: 125 12 2 − 6 3 2 3 ≈ 16.456
Divide Expression 1 by Expression 3: 3 6 2 3 = 12 2 ≈ 0.1179
Explanation
Understanding the Problem We are given three expressions:
Expression 1: 4 6 3 8
Expression 2: 5 3 12 2 − 6 3
Expression 3: 3 6
The question asks for the 'quarient', which I assume is a typo and meant to be 'quotient'. However, it is unclear what is divided by what. I will calculate the quotient of Expression 1 divided by Expression 2, and then the quotient of Expression 1 divided by Expression 3.
Simplifying Expression 1 First, let's simplify Expression 1: 4 6 3 8 = 4 6 3 4 ⋅ 2 = 4 6 3 ⋅ 2 2 = 4 6 6 2 = 2 6 3 2 = 2 2 ⋅ 3 3 2 = 2 3 3 = 2 ⋅ 3 3 3 = 2 3 So, Expression 1 simplifies to 2 3 .
Simplifying Expression 2 Next, let's simplify Expression 2: 5 3 12 2 − 6 3 = 125 12 2 − 6 3 Expression 2 is already in a relatively simplified form.
Dividing Expression 1 by Expression 2 Now, let's divide Expression 1 by Expression 2: 125 12 2 − 6 3 2 3 = 2 3 ⋅ 12 2 − 6 3 125 = 2 ( 12 2 − 6 3 ) 125 3 = 24 2 − 12 3 125 3 Using a calculator, the value of this expression is approximately 16.456.
Dividing Expression 1 by Expression 3 Now, let's divide Expression 1 by Expression 3: 3 6 2 3 = 2 3 ⋅ 3 6 1 = 6 6 3 = 6 2 ⋅ 3 3 = 6 2 3 3 = 6 2 1 = 6 ⋅ 2 2 = 12 2 Using a calculator, the value of this expression is approximately 0.1179.
Final Answer The quotient of Expression 1 divided by Expression 2 is approximately 16.456, and the quotient of Expression 1 divided by Expression 3 is approximately 0.1179.
Examples
Understanding quotients helps in various real-life scenarios, such as calculating ratios in recipes (e.g., the ratio of flour to sugar), determining unit prices when shopping (e.g., price per ounce), or analyzing financial ratios in business (e.g., debt-to-equity ratio). In essence, quotients enable us to compare quantities and make informed decisions based on relative values.
