What is the vlaue of `(3 ^(5) xx 27^(3))/(81^(3))` ?

Answer

To solve the expression \((3^{5} \times 27^{3})/(81^{3})\), we will simplify each term using the laws of exponents step by step.

### Step 1: Rewrite the bases in terms of 3
First, we need to express \(27\) and \(81\) as powers of \(3\):
- \(27 = 3^{3}\)
- \(81 = 3^{4}\)

Now we can rewrite the expression:
\[ (3^{5} \times (3^{3})^{3})/( (3^{4})^{3}) \]

### Step 2: Simplify the powers
Using the power of a power rule \((a^{m})^{n} = a^{m \cdot n}\), we can simplify:
\[ (3^{5} \times 3^{9})/(3^{12}) \]

### Step 3: Combine the terms in the numerator
Now, we can combine the terms in the numerator using the product of powers rule \(a^{m} \times a^{n} = a^{m+n}\):
\[ 3^{5 + 9}/3^{12} = 3^{14}/3^{12} \]

### Step 4: Simplify the division
Now, we will simplify the expression using the quotient of powers rule \(a^{m}/a^{n} = a^{m-n}\):
\[ 3^{14 - 12} = 3^{2} \]

### Step 5: Calculate the final value
Now we can calculate \(3^{2}\):
\[ 3^{2} = 9 \]

Thus, the value of \((3^{5} \times 27^{3})/(81^{3})\) is \(\boxed{9}\).