`(3)1/4+(4)1/6+?1/4=10`

Options

A. (2)1/6
B. (4)1/3
C. (1)1/3
D. (2)1/3

Answer

To solve the equation \( (3)^{1/4} + (4)^{1/6} + ?^{1/4} = 10 \), we will follow these steps:

### Step 1: Rewrite the equation
Let's rewrite the equation by substituting the question mark with \( x \):
\[ 3^{1/4} + 4^{1/6} + x^{1/4} = 10 \]

### Step 2: Calculate \( 3^{1/4} \) and \( 4^{1/6} \)
We need to calculate the values of \( 3^{1/4} \) and \( 4^{1/6} \):
- \( 3^{1/4} \) is the fourth root of 3.
- \( 4^{1/6} \) is the sixth root of 4.

Using approximate values:
- \( 3^{1/4} \approx 1.316 \)
- \( 4^{1/6} \approx 1.414 \)

### Step 3: Substitute the values into the equation
Now substitute these values back into the equation:
\[ 1.316 + 1.414 + x^{1/4} = 10 \]

### Step 4: Combine the constants
Combine the constants on the left side:
\[ 2.730 + x^{1/4} = 10 \]

### Step 5: Isolate \( x^{1/4} \)
Now, isolate \( x^{1/4} \):
\[ x^{1/4} = 10 - 2.730 \]
\[ x^{1/4} = 7.270 \]

### Step 6: Raise both sides to the power of 4
To solve for \( x \), raise both sides to the power of 4:
\[ x = (7.270)^4 \]

### Step 7: Calculate \( x \)
Calculating \( (7.270)^4 \):
\[ x \approx 2755.8 \]

### Step 8: Final value of \( x \)
Thus, the value of \( x \) is approximately:
\[ x \approx 2756 \]

### Conclusion
The question mark \( ? \) is approximately equal to \( 2756 \).

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