`root(3)((19)/(513))` is equal to

Options

A. a) `(1)/(9)`
B. b) `(1)/(3)`
C. c) `(1)/(sqrt(27))`
D. d) `(1)/(sqrt(3))`

Answer

To solve the problem \( \sqrt[3]{\frac{19}{513}} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt[3]{\frac{19}{513}} \]

### Step 2: Factor the denominator
Next, we can factor the denominator \( 513 \). We find that:
\[ 513 = 19 \times 27 \]
So we can rewrite the expression as:
\[ \sqrt[3]{\frac{19}{19 \times 27}} \]

### Step 3: Simplify the fraction
Now, we can simplify the fraction:
\[ \frac{19}{19 \times 27} = \frac{1}{27} \]
Thus, the expression becomes:
\[ \sqrt[3]{\frac{1}{27}} \]

### Step 4: Apply the cube root
The cube root of \( \frac{1}{27} \) can be expressed as:
\[ \sqrt[3]{\frac{1}{27}} = \frac{1}{\sqrt[3]{27}} \]
Since \( 27 = 3^3 \), we have:
\[ \sqrt[3]{27} = 3 \]
So, we can write:
\[ \frac{1}{\sqrt[3]{27}} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{19}{513}} \) is:
\[ \frac{1}{3} \]

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