The number of solutions of the equation `root3((1+x))+root3((8-x))=3` is equal to

Options

A. two
B. one
C. zero
D. none of these

Answer

To solve the equation \( \sqrt[3]{1+x} + \sqrt[3]{8-x} = 3 \), we will follow these steps:

### Step 1: Isolate the cube roots
We start with the equation:
\[ \sqrt[3]{1+x} + \sqrt[3]{8-x} = 3 \]

### Step 2: Cube both sides
Cubing both sides gives us:
\[ \left(\sqrt[3]{1+x} + \sqrt[3]{8-x}\right)^3 = 3^3 \]
Using the identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \), we have:
\[ \sqrt[3]{(1+x)}^3 + \sqrt[3]{(8-x)}^3 + 3\sqrt[3]{(1+x)}\sqrt[3]{(8-x)}(3) = 27 \]
This simplifies to:
\[ (1+x) + (8-x) + 9\sqrt[3]{(1+x)(8-x)} = 27 \]

### Step 3: Simplify the equation
Combining the terms:
\[ 9 + 9\sqrt[3]{(1+x)(8-x)} = 27 \]
Subtracting 9 from both sides:
\[ 9\sqrt[3]{(1+x)(8-x)} = 18 \]
Dividing both sides by 9:
\[ \sqrt[3]{(1+x)(8-x)} = 2 \]

### Step 4: Cube both sides again
Cubing both sides again gives us:
\[ (1+x)(8-x) = 2^3 \]
This simplifies to:
\[ (1+x)(8-x) = 8 \]

### Step 5: Expand the left side
Expanding the left side:
\[ 8 - x + 8x - x^2 = 8 \]
This simplifies to:
\[ -x^2 + 7x = 0 \]

### Step 6: Factor the equation
Factoring out \( x \):
\[ x(7-x) = 0 \]
Setting each factor to zero gives us:
\[ x = 0 \quad \text{or} \quad 7 - x = 0 \Rightarrow x = 7 \]

### Step 7: Determine the number of solutions
Thus, we have two solutions: \( x = 0 \) and \( x = 7 \).

### Conclusion
The number of solutions to the equation \( \sqrt[3]{1+x} + \sqrt[3]{8-x} = 3 \) is **2**.

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