simplify `81^((1)/(4)) xx 9^((3)/(2)) xx 27^(-(4)/(3))`= ____

Options

A. 1
B. 3
C. 9
D. `(1)/(3)`

Answer

To simplify the expression \( 81^{\frac{1}{4}} \times 9^{\frac{3}{2}} \times 27^{-\frac{4}{3}} \), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 3
- \( 81 = 3^4 \)
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

### Step 2: Substitute these values into the expression
Now we can rewrite the expression:
\[ 81^{\frac{1}{4}} = (3^4)^{\frac{1}{4}}, \quad 9^{\frac{3}{2}} = (3^2)^{\frac{3}{2}}, \quad 27^{-\frac{4}{3}} = (3^3)^{-\frac{4}{3}} \]

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

### Step 4: Combine the terms
Now we can combine all the terms:
\[ 3^1 \times 3^3 \times 3^{-4} \]

### Step 5: Use the property of exponents to add the exponents
Using the property \(a^m \times a^n = a^{m+n}\):
\[ 3^{1 + 3 - 4} = 3^{0} \]

### Step 6: Simplify \(3^0\)
We know that any non-zero number raised to the power of 0 is 1:
\[ 3^0 = 1 \]

### Final Answer
Thus, the simplified expression is:
\[ \boxed{1} \]