What is the value of `[(( x^(y +1))^((y^(2))/(y^(2) -1)))^(1 -(1)/(y)) ] `?

Options

A. xy
B. `x^(y)`
C. `y^(x)`
D. `y//x`

Answer

To solve the expression \[ \left[ \left( x^{(y + 1)} \right)^{\left( \frac{y^2}{y^2 - 1} \right)} \right]^{\left( 1 - \frac{1}{y} \right)} \], we can follow these steps:

### Step 1: Apply the Power of a Power Rule
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can rewrite the expression:

\[ = x^{(y + 1) \cdot \left( \frac{y^2}{y^2 - 1} \right) \cdot \left( 1 - \frac{1}{y} \right)} \]

### Step 2: Simplify the Exponent
Now, we need to simplify the exponent:

\[ = (y + 1) \cdot \frac{y^2}{y^2 - 1} \cdot \left( 1 - \frac{1}{y} \right) \]

First, simplify \(1 - \frac{1}{y}\):

\[ 1 - \frac{1}{y} = \frac{y - 1}{y} \]

Substituting this back into the exponent gives:

\[ = (y + 1) \cdot \frac{y^2}{y^2 - 1} \cdot \frac{y - 1}{y} \]

### Step 3: Combine Terms
Now we can simplify the expression further:

\[ = \frac{(y + 1)(y - 1)y^2}{y(y^2 - 1)} \]

The \(y\) in the numerator and denominator cancels out:

\[ = \frac{(y + 1)(y - 1)y}{y^2 - 1} \]

### Step 4: Recognize the Difference of Squares
Notice that \(y^2 - 1\) can be factored as \((y + 1)(y - 1)\):

\[ = \frac{(y + 1)(y - 1)y}{(y + 1)(y - 1)} \]

### Step 5: Cancel Common Factors
The \((y + 1)(y - 1)\) terms cancel out:

\[ = y \]

### Final Step: Write the Final Expression
Thus, we have:

\[ x^y \]

### Conclusion
The value of the original expression is:

\[ \boxed{x^y} \]