If `root(3)(a)=root(3)(26)+root(3)(7)+root(3)(63)`, then-

Options

A. alt 729 but agt 216
B. alt 216
C. agt 729
D. agt 729

Answer

To solve the equation \( \sqrt[3]{a} = \sqrt[3]{26} + \sqrt[3]{7} + \sqrt[3]{63} \), we will follow these steps:

### Step 1: Simplify the equation
We know that \( \sqrt[3]{63} \) can be simplified. Since \( 63 = 3^2 \times 7 \), we can rewrite it as:
\[ \sqrt[3]{63} = \sqrt[3]{3^2 \times 7} = \sqrt[3]{9} \cdot \sqrt[3]{7} \]

### Step 2: Substitute back into the equation
Now, substituting this back into the original equation gives us:
\[ \sqrt[3]{a} = \sqrt[3]{26} + \sqrt[3]{7} + \sqrt[3]{9} \cdot \sqrt[3]{7} \]

### Step 3: Combine like terms
Notice that \( \sqrt[3]{7} \) is common in two terms. We can factor it out:
\[ \sqrt[3]{a} = \sqrt[3]{26} + \sqrt[3]{7}(1 + \sqrt[3]{9}) \]

### Step 4: Calculate the approximate values
Now we can calculate approximate values:
- \( \sqrt[3]{26} \approx 2.96 \)
- \( \sqrt[3]{7} \approx 1.91 \)
- \( \sqrt[3]{9} \approx 2.08 \)

Calculating \( 1 + \sqrt[3]{9} \):
\[ 1 + \sqrt[3]{9} \approx 1 + 2.08 = 3.08 \]

Now substituting these values back:
\[ \sqrt[3]{a} \approx 2.96 + 1.91 \cdot 3.08 \]
Calculating \( 1.91 \cdot 3.08 \):
\[ 1.91 \cdot 3.08 \approx 5.88 \]
Thus,
\[ \sqrt[3]{a} \approx 2.96 + 5.88 \approx 8.84 \]

### Step 5: Cube both sides to find \( a \)
Now we cube both sides to find \( a \):
\[ a \approx (8.84)^3 \]
Calculating \( (8.84)^3 \):
\[ (8.84)^3 \approx 693.5 \]

### Step 6: Conclusion
Thus, we can conclude that:
\[ a \approx 693.5 \]