If `sin ^(-1) x + sin ^(-1) y + sin ^(-1) z = (3pi)/(2)` then the value of `x ^(100) + y ^(100) + z ^(100) - (3)/( x ^(101) + y^(101) + z ^(101)) ` is

Answer

To solve the problem, we start with the given equation:

\[ \sin^{-1} x + \sin^{-1} y + \sin^{-1} z = \frac{3\pi}{2} \]

### Step 1: Analyze the equation
The maximum value of \(\sin^{-1} t\) for any real number \(t\) is \(\frac{\pi}{2}\), which occurs when \(t = 1\). Since the sum of three arcsine functions equals \(\frac{3\pi}{2}\), each term must equal \(\frac{\pi}{2}\) for the equation to hold true. Therefore, we conclude:

\[ \sin^{-1} x = \frac{\pi}{2}, \quad \sin^{-1} y = \frac{\pi}{2}, \quad \sin^{-1} z = \frac{\pi}{2} \]

### Step 2: Solve for \(x\), \(y\), and \(z\)
From \(\sin^{-1} x = \frac{\pi}{2}\), we can deduce:

\[ x = \sin\left(\frac{\pi}{2}\right) = 1 \]
Similarly, we find:

\[ y = 1, \quad z = 1 \]

### Step 3: Substitute values into the expression
Now we need to evaluate the expression:

\[ x^{100} + y^{100} + z^{100} - \frac{3}{x^{101} + y^{101} + z^{101}} \]

Substituting \(x = 1\), \(y = 1\), and \(z = 1\):

\[ 1^{100} + 1^{100} + 1^{100} - \frac{3}{1^{101} + 1^{101} + 1^{101}} \]

### Step 4: Simplify the expression
Calculating each term:

\[ 1^{100} = 1, \quad 1^{101} = 1 \]

Thus, we have:

\[ 1 + 1 + 1 - \frac{3}{1 + 1 + 1} = 3 - \frac{3}{3} \]

This simplifies to:

\[ 3 - 1 = 2 \]

### Final Answer
The value of the expression is:

\[ \boxed{2} \]