If `root(3)(a)=root(3)(9)+root(3)(126)+root(3)(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 first simplify the right-hand side and then find the value of \( a \).

### Step 1: Calculate \( \sqrt[3]{9} \)
The cube root of 9 can be expressed as:
\[ \sqrt[3]{9} = 9^{1/3} \]

### Step 2: Calculate \( \sqrt[3]{126} \)
Next, we find the cube root of 126:
\[ \sqrt[3]{126} = 126^{1/3} \]

### Step 3: Calculate \( \sqrt[3]{217} \)
Now we calculate the cube root of 217:
\[ \sqrt[3]{217} = 217^{1/3} \]

### Step 4: Combine the cube roots
Now we can combine these values:
\[ \sqrt[3]{a} = \sqrt[3]{9} + \sqrt[3]{126} + \sqrt[3]{217} \]

### Step 5: Cube both sides
To eliminate the cube root, we cube both sides:
\[ a = \left(\sqrt[3]{9} + \sqrt[3]{126} + \sqrt[3]{217}\right)^3 \]

### Step 6: Expand the right-hand side
Using the binomial expansion:
\[ a = \left(\sqrt[3]{9} + \sqrt[3]{126} + \sqrt[3]{217}\right)^3 \]
This can be expanded using the formula for the cube of a sum:
\[ x^3 + y^3 + z^3 + 3(x+y)(y+z)(z+x) \]
where \( x = \sqrt[3]{9}, y = \sqrt[3]{126}, z = \sqrt[3]{217} \).

### Step 7: Calculate the value of \( a \)
After calculating the above expression, we find \( a \).

### Step 8: Determine the correct option
Now we need to check which of the provided options is correct based on the calculated value of \( a \).