If `(125)^(2//3) xx (625)^(-1//4) = 5^(x)`, then the value of x is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( (125)^{\frac{2}{3}} \times (625)^{-\frac{1}{4}} = 5^x \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 5
We know that:
- \( 125 = 5^3 \)
- \( 625 = 5^4 \)

So we can rewrite the equation as:
\[ (5^3)^{\frac{2}{3}} \times (5^4)^{-\frac{1}{4}} = 5^x \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify the left-hand side:
\[ 5^{3 \cdot \frac{2}{3}} \times 5^{4 \cdot -\frac{1}{4}} = 5^x \]

### Step 3: Simplify the exponents
Calculating the exponents:
- For \( 5^{3 \cdot \frac{2}{3}} \):
\[ 3 \cdot \frac{2}{3} = 2 \]
- For \( 5^{4 \cdot -\frac{1}{4}} \):
\[ 4 \cdot -\frac{1}{4} = -1 \]

So now we have:
\[ 5^2 \times 5^{-1} = 5^x \]

### Step 4: Combine the exponents
Using the property \( a^m \times a^n = a^{m+n} \):
\[ 5^{2 + (-1)} = 5^x \]

This simplifies to:
\[ 5^{2 - 1} = 5^x \]
\[ 5^1 = 5^x \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ 1 = x \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{1} \]