Solve for x : `(81)^((3)/(4))-((1)/(32))^(-(2)/(5))+x((1)/(2))^(-1).2^(0) = 27`

Answer

To solve the equation \( (81)^{\frac{3}{4}} - \left(\frac{1}{32}\right)^{-\frac{2}{5}} + x\left(\frac{1}{2}\right)^{-1} \cdot 2^{0} = 27 \), we will follow these steps:

### Step 1: Rewrite the terms using exponents
First, we will express the numbers in terms of their base and exponent forms.

- \( 81 = 3^4 \), so \( (81)^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^{4 \cdot \frac{3}{4}} = 3^3 \).
- \( \frac{1}{32} = 32^{-1} = (2^5)^{-1} = 2^{-5} \), thus \( \left(\frac{1}{32}\right)^{-\frac{2}{5}} = (2^{-5})^{-\frac{2}{5}} = 2^{5 \cdot \frac{2}{5}} = 2^2 \).
- \( \left(\frac{1}{2}\right)^{-1} = 2^1 \) and \( 2^0 = 1 \).

So we can rewrite the equation as:
\[ 3^3 - 2^2 + x \cdot 2^1 \cdot 1 = 27 \]

### Step 2: Simplify the equation
Now we can simplify the equation:
\[ 3^3 - 2^2 + 2x = 27 \]
Calculating \( 3^3 \) and \( 2^2 \):
\[ 27 - 4 + 2x = 27 \]

### Step 3: Solve for \( x \)
Next, we can simplify further:
\[ 27 - 4 = 23 \]
So the equation becomes:
\[ 23 + 2x = 27 \]

Now, isolate \( 2x \):
\[ 2x = 27 - 23 \]
\[ 2x = 4 \]

Finally, divide by 2:
\[ x = \frac{4}{2} = 2 \]

### Final Answer
Thus, the value of \( x \) is \( 2 \).

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