`(32/243)^(3//5)` = ?

Options

A. `27/8`
B. `27/7`
C. `27/6`
D. `27/2`

Answer

To solve the expression \((32/243)^{3/5}\), we will follow these steps:

### Step 1: Factor the Base
First, we need to express \(32\) and \(243\) in terms of their prime factors.

- \(32 = 2^5\) (since \(2 \times 2 \times 2 \times 2 \times 2 = 32\))
- \(243 = 3^5\) (since \(3 \times 3 \times 3 \times 3 \times 3 = 243\))

Thus, we can rewrite the fraction:
\[ \frac{32}{243} = \frac{2^5}{3^5} \]

### Step 2: Apply the Power
Now we substitute the factored form into the expression:
\[ (32/243)^{3/5} = \left(\frac{2^5}{3^5}\right)^{3/5} \]

### Step 3: Use the Power of a Quotient Rule
Using the property of exponents that states \((a/b)^m = a^m/b^m\), we can separate the powers:
\[ \left(\frac{2^5}{3^5}\right)^{3/5} = \frac{(2^5)^{3/5}}{(3^5)^{3/5}} \]

### Step 4: Simplify the Exponents
Now we simplify the exponents using the property \((a^m)^n = a^{m \cdot n}\):
\[ (2^5)^{3/5} = 2^{5 \cdot (3/5)} = 2^3 \]
\[ (3^5)^{3/5} = 3^{5 \cdot (3/5)} = 3^3 \]

### Step 5: Substitute Back
Now we can substitute back into our expression:
\[ \frac{(2^3)}{(3^3)} = \frac{2^3}{3^3} \]

### Step 6: Calculate the Values
Now we calculate \(2^3\) and \(3^3\):
\[ 2^3 = 8 \quad \text{and} \quad 3^3 = 27 \]

Thus, we have:
\[ \frac{2^3}{3^3} = \frac{8}{27} \]

### Final Answer
Therefore, the value of \((32/243)^{3/5}\) is:
\[ \frac{8}{27} \]

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