The value of `(343/1331)^(1//3)` is-

Options

A. `5/11`
B. `7/5`
C. `7/11`
D. `7/10`

Answer

To solve the expression \((\frac{343}{1331})^{\frac{1}{3}}\), we can follow these steps:

### Step 1: Identify the cubes
First, we recognize that both 343 and 1331 can be expressed as cubes of integers:
- \(343 = 7^3\)
- \(1331 = 11^3\)

### Step 2: Rewrite the expression
Using the cube representations, we can rewrite the expression:
\[ \frac{343}{1331} = \frac{7^3}{11^3} \]

### Step 3: Apply the power of a quotient rule
Now we can apply the power of a quotient rule, which states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\):
\[ \left(\frac{7^3}{11^3}\right)^{\frac{1}{3}} = \frac{(7^3)^{\frac{1}{3}}}{(11^3)^{\frac{1}{3}}} \]

### Step 4: Simplify the powers
Using the power of a power rule \((a^m)^n = a^{m \cdot n}\):
\[ (7^3)^{\frac{1}{3}} = 7^{3 \cdot \frac{1}{3}} = 7^1 = 7 \]
\[ (11^3)^{\frac{1}{3}} = 11^{3 \cdot \frac{1}{3}} = 11^1 = 11 \]

### Step 5: Combine the results
Now we can combine the results from the simplifications:
\[ \frac{(7^3)^{\frac{1}{3}}}{(11^3)^{\frac{1}{3}}} = \frac{7}{11} \]

### Final Answer
Thus, the value of \((\frac{343}{1331})^{\frac{1}{3}}\) is:
\[ \frac{7}{11} \]

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