If `x= (8^((2)/(3)).32^(-(2)/(5)))`, then `x^(-5)`= ____

Options

A. `(1)/(32)`
B. `-1`
C. 1
D. `-5`

Answer

To solve the problem, we start with the expression given for \( x \):

\[ x = 8^{\frac{2}{3}} \cdot 32^{-\frac{2}{5}} \]

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \( 8 = 2^3 \)
- \( 32 = 2^5 \)

Now, we can rewrite \( x \):

\[ x = (2^3)^{\frac{2}{3}} \cdot (2^5)^{-\frac{2}{5}} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify both terms:

\[ x = 2^{3 \cdot \frac{2}{3}} \cdot 2^{5 \cdot -\frac{2}{5}} \]

### Step 3: Simplify the exponents
Calculating the exponents gives us:

\[ 3 \cdot \frac{2}{3} = 2 \quad \text{and} \quad 5 \cdot -\frac{2}{5} = -2 \]

Thus, we have:

\[ x = 2^2 \cdot 2^{-2} \]

### Step 4: Combine the exponents
Using the property \( a^m \cdot a^n = a^{m+n} \):

\[ x = 2^{2 + (-2)} = 2^0 \]

### Step 5: Simplify \( 2^0 \)
We know that any non-zero number raised to the power of 0 is 1:

\[ x = 1 \]

### Step 6: Calculate \( x^{-5} \)
Now we need to find \( x^{-5} \):

\[ x^{-5} = 1^{-5} \]

Since \( 1 \) raised to any power is still \( 1 \):

\[ x^{-5} = 1 \]

### Final Answer:
Thus, the value of \( x^{-5} \) is:

\[ \boxed{1} \]