The value of `(((243)^(5))^(4))/(((32)^(4))^(5))` is

Options

A. `(3)/(2)`
B. `(3/2)^(100)`
C. `(1)/(2^(3)xx3^(3))`
D. `(1)/(2^(4)xx3^(4))`

Answer

To solve the expression \(\frac{((243)^{5})^{4}}{((32)^{4})^{5}}\), we will follow these steps:

### Step 1: Simplify the Exponents
Using the exponent rule \((a^m)^n = a^{m \cdot n}\), we can simplify the expression:
\[ \frac{(243^{5 \cdot 4})}{(32^{4 \cdot 5})} = \frac{(243^{20})}{(32^{20})} \]

### Step 2: Rewrite the Base Numbers
Next, we will express 243 and 32 as powers of their prime factors:
- \(243 = 3^5\) (since \(3^5 = 243\))
- \(32 = 2^5\) (since \(2^5 = 32\))

Now we can substitute these values into our expression:
\[ \frac{((3^5)^{20})}{((2^5)^{20})} \]

### Step 3: Apply the Exponent Rule Again
Using the same exponent rule, we can simplify further:
\[ \frac{3^{5 \cdot 20}}{2^{5 \cdot 20}} = \frac{3^{100}}{2^{100}} \]

### Step 4: Combine the Expression
We can combine the expression using the property \(\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m\):
\[ \frac{3^{100}}{2^{100}} = \left(\frac{3}{2}\right)^{100} \]

### Final Result
Thus, the value of the original expression is:
\[ \left(\frac{3}{2}\right)^{100} \]