Find `x^(x)," if "(64)/(27)= (4/3)^(x)`.

Answer

To solve the equation \( \frac{64}{27} = \left(\frac{4}{3}\right)^x \) and find \( x^x \), we can follow these steps:

### Step 1: Rewrite the numbers in terms of their prime factors.
We know that:
- \( 64 = 4^3 \) (since \( 4 \times 4 \times 4 = 64 \))
- \( 27 = 3^3 \) (since \( 3 \times 3 \times 3 = 27 \))

So we can rewrite the left-hand side:
\[ \frac{64}{27} = \frac{4^3}{3^3} \]

### Step 2: Simplify the left-hand side.
Using the property of exponents, we can simplify:
\[ \frac{4^3}{3^3} = \left(\frac{4}{3}\right)^3 \]

### Step 3: Set the two sides equal.
Now we have:
\[ \left(\frac{4}{3}\right)^3 = \left(\frac{4}{3}\right)^x \]

### Step 4: Equate the exponents.
Since the bases are the same, we can equate the exponents:
\[ 3 = x \]

### Step 5: Find \( x^x \).
Now that we have found \( x = 3 \), we can find \( x^x \):
\[ x^x = 3^3 \]

### Step 6: Calculate \( 3^3 \).
Calculating \( 3^3 \):
\[ 3^3 = 27 \]

### Final Answer:
Thus, the value of \( x^x \) is \( 27 \).

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