What is $\frac{\sqrt[3]{3}}{\sqrt[3]{2 x}}$ in simplest form?
A. $\frac{\sqrt[3]{12 x^2}}{2 x}$
B. $\frac{\sqrt[3]{6 x}}{2 x}$
C. $\frac{\sqrt[3]{3}}{2 x}$
Answer (2)
The expression 3 2 x 3 3 simplifies to 2 x 3 12 x 2 after rationalizing the denominator. Therefore, the correct answer is option A: 2 x 3 12 x 2 .
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Rationalize the denominator by multiplying both the numerator and denominator by 3 ( 2 x ) 2 .
Simplify the expression: 3 2 x 3 3 ⋅ 3 ( 2 x ) 2 3 ( 2 x ) 2 = 2 x 3 12 x 2 .
The simplest form of the given expression is 2 x 3 12 x 2 .
The final answer is 2 x 3 12 x 2 .
Explanation
Understanding the Problem We are given the expression 3 2 x 3 3 and we want to simplify it. To do this, we will rationalize the denominator.
Rationalizing the Denominator To rationalize the denominator, we need to multiply both the numerator and the denominator by a factor that will eliminate the cube root in the denominator. In this case, we can multiply by 3 ( 2 x ) 2 .
Performing the Multiplication Multiplying the numerator and denominator by 3 ( 2 x ) 2 gives us: 3 2 x 3 3 ⋅ 3 ( 2 x ) 2 3 ( 2 x ) 2 = 3 2 x ⋅ 3 ( 2 x ) 2 3 3 ⋅ 3 ( 2 x ) 2 = 3 ( 2 x ) 3 3 3 ⋅ ( 2 x ) 2 = 2 x 3 3 ⋅ 4 x 2 = 2 x 3 12 x 2 .
Comparing with Options Now we compare our simplified expression 2 x 3 12 x 2 with the given options. We see that it matches the first option.
Final Answer Therefore, the simplest form of the given expression is 2 x 3 12 x 2 .
Examples
Simplifying radical expressions is useful in various fields, such as engineering and physics, where complex calculations often involve radicals. For example, when calculating the impedance of an electrical circuit or determining the energy levels of a quantum mechanical system, simplifying radical expressions can make the calculations more manageable and easier to interpret. In architecture, simplifying radical expressions can help in calculating dimensions and proportions accurately, ensuring structural integrity and aesthetic appeal.
