Value of `((1024)/(243))^((3)/(5))` is _____

Options

A. `(128)/(27)`
B. `(32)/(27)`
C. `(64)/(27)`
D. `(32)/(9)`

Explanation

To find the value of \(\left(\frac{1024}{243}\right)^{\frac{3}{5}}\), we can simplify the expression step by step.

### Step 1: Express 1024 and 243 as powers of their prime factors.
- **1024** can be expressed as \(2^{10}\) because \(2^{10} = 1024\).
- **243** can be expressed as \(3^5\) because \(3^5 = 243\).

### Step 2: Rewrite the expression using these powers.
Now we can rewrite the original expression:
\[ \left(\frac{1024}{243}\right)^{\frac{3}{5}} = \left(\frac{2^{10}}{3^5}\right)^{\frac{3}{5}} \]

### Step 3: Apply the power of a quotient rule.
Using the property of exponents \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\), we can simplify:
\[ \left(\frac{2^{10}}{3^5}\right)^{\frac{3}{5}} = \frac{(2^{10})^{\frac{3}{5}}}{(3^5)^{\frac{3}{5}}} \]

### Step 4: Simplify the powers.
Now we simplify the powers:
\[ (2^{10})^{\frac{3}{5}} = 2^{10 \cdot \frac{3}{5}} = 2^6 \]
\[ (3^5)^{\frac{3}{5}} = 3^{5 \cdot \frac{3}{5}} = 3^3 \]

### Step 5: Rewrite the expression with the simplified powers.
Now we have:
\[ \frac{2^6}{3^3} \]

### Step 6: Calculate the values of the powers.
Calculating the powers:
\[ 2^6 = 64 \quad \text{and} \quad 3^3 = 27 \]

### Step 7: Write the final answer.
Thus, the expression simplifies to:
\[ \frac{64}{27} \]

### Final Answer:
The value of \(\left(\frac{1024}{243}\right)^{\frac{3}{5}}\) is \(\frac{64}{27}\).

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