`root3(a)=root3(9)+root3(126)+root3(217)`, then which of the following is correct ?

Options

A. `a=2197`
B. `a gt 2197`
C. `a lt 2197`
D. `a lt 1728`

Answer

To solve the equation \( \sqrt[3]{a} = \sqrt[3]{9} + \sqrt[3]{126} + \sqrt[3]{217} \), we will follow these steps:

### Step 1: Estimate the values of the cube roots
First, we need to estimate the values of \( \sqrt[3]{9} \), \( \sqrt[3]{126} \), and \( \sqrt[3]{217} \).

- \( \sqrt[3]{9} \) is slightly greater than \( 2 \) because \( 2^3 = 8 \).
- \( \sqrt[3]{126} \) is slightly greater than \( 5 \) because \( 5^3 = 125 \).
- \( \sqrt[3]{217} \) is slightly greater than \( 6 \) because \( 6^3 = 216 \).

### Step 2: Add the estimated values
Now we can add these estimates together:
\[ \sqrt[3]{9} + \sqrt[3]{126} + \sqrt[3]{217} > 2 + 5 + 6 = 13 \]
Thus, we have:
\[ \sqrt[3]{a} > 13 \]

### Step 3: Cube both sides
Next, we cube both sides of the inequality:
\[ a > 13^3 \]
Calculating \( 13^3 \):
\[ 13^3 = 2197 \]
So, we have:
\[ a > 2197 \]

### Step 4: Analyze the options
Now we can analyze the options given:
- \( a = 2197 \)
- \( a > 2119 \)
- \( a < 1728 \)

From our calculation, we found that \( a > 2197 \). Therefore, the correct option is:
- \( a > 2119 \) (since 2197 is greater than 2119).

### Conclusion
The correct answer is that \( a \) is greater than 2119.

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