If B varies inversely as c$^{\frac{1}{3}}$ and C = 27 when B = 2, find the value of the constant of proportionality K.

1. Analyze the question

We are given that $B$ varies inversely as $C^{\frac{1}{3}}$. We are provided with a pair of values ($C = 27$ and $B = 2$) and asked to find the constant of proportionality, $k$.

2. Explain the concept

In mathematics, when a variable $B$ varies inversely as another variable $X$, it is written as:

where $k$ is the constant of proportionality.

In this specific problem, $X = C^{\frac{1}{3}}$, so our equation becomes:

3. Show the solution

Step 1: Set up the equation.

Step 2: Substitute the given values into the equation. We are told $B = 2$ and $C = 27$.

Step 3: Simplify the cube root. Recall that $27^{\frac{1}{3}}$ is the same as $\sqrt[3]{27}$. Since $3 \times 3 \times 3 = 27$, then $\sqrt[3]{27} = 3$. So, the equation becomes:

Step 4: Solve for $k$. Multiply both sides by 3:

4. Identify the answer

Comparing our result ($k = 6$) with the given options, the correct choice is B.

5. Tip

Whenever you see a fractional exponent like $x^{\frac{1}{n}}$, remember it is just a fancy way of writing the $n^{th}$ root ($\sqrt[n]{x}$). If you are dealing with large numbers, try to express them as powers of smaller numbers (e.g., $27 = 3^3$) to make the calculation much easier!