Which of the following is not equal to `((-5)/4)^(4)`?

Options

A. `((-5)^(4))/(4^(4))`
B. `(5^(4))/((-4)^(4))`
C. `-(5^(4))/(4^(4))`
D. `(-5/4)xx(-5/4)xx(-5/4)xx(-5/4)`

Answer

To solve the problem of determining which of the following options is not equal to \((-5/4)^4\), we will evaluate each option step by step.

### Step 1: Understand the expression
The expression we are working with is \((-5/4)^4\). This means we are raising the fraction \(-5/4\) to the power of 4.

### Step 2: Evaluate the first option
**Option 1:** \((-5/4)^4\)

This option is clearly equal to \((-5/4)^4\) since it is the same expression.

### Step 3: Evaluate the second option
**Option 2:** \(\frac{5^4}{(-4)^4}\)

Using the property of exponents that states \(\frac{a^m}{b^m} = \frac{a}{b}^m\), we can rewrite this as:

\[ \frac{5^4}{(-4)^4} = \frac{5^4}{4^4} \cdot \frac{1}{(-1)^4} = \frac{5^4}{4^4} = \left(\frac{5}{4}\right)^4 \]

Since \((-1)^4 = 1\), this option simplifies to \(\left(\frac{5}{4}\right)^4\), which is not equal to \((-5/4)^4\).

### Step 4: Evaluate the third option
**Option 3:** \(\frac{(-5)^4}{4^4}\)

This can be simplified as follows:

\[ \frac{(-5)^4}{4^4} = \left(\frac{-5}{4}\right)^4 \]

This is equal to \((-5/4)^4\).

### Step 5: Evaluate the fourth option
**Option 4:** \((-5/4) \times (-5/4) \times (-5/4) \times (-5/4)\)

This can be rewritten as:

\[ \left(-\frac{5}{4}\right)^4 \]

This is also equal to \((-5/4)^4\).

### Conclusion
From our evaluations:
- Option 1 is equal to \((-5/4)^4\).
- Option 2 is not equal to \((-5/4)^4\).
- Option 3 is equal to \((-5/4)^4\).
- Option 4 is equal to \((-5/4)^4\).

Thus, the option that is **not equal** to \((-5/4)^4\) is **Option 2: \(\frac{5^4}{(-4)^4}\)**.