`4xx(256)^((-1)/4) div (243)^(1/5)=`

Options

A. `1/3`
B. `4/3`
C. `3/4`
D. 1

Explanation

To solve the expression \( 4 \times (256)^{-\frac{1}{4}} \div (243)^{\frac{1}{5}} \), we will follow these steps:

### Step 1: Rewrite the numbers in terms of their prime factors.
- \( 4 = 2^2 \)
- \( 256 = 2^8 \) (since \( 256 = 2^8 \))
- \( 243 = 3^5 \) (since \( 243 = 3^5 \))

### Step 2: Substitute these values into the expression.
The expression now becomes:
\[ 2^2 \times (2^8)^{-\frac{1}{4}} \div (3^5)^{\frac{1}{5}} \]

### Step 3: Simplify the powers using the power of a power property.
Using the property \( (a^m)^n = a^{m \cdot n} \):
- \( (2^8)^{-\frac{1}{4}} = 2^{8 \cdot -\frac{1}{4}} = 2^{-2} \)
- \( (3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}} = 3^1 = 3 \)

Now the expression becomes:
\[ 2^2 \times 2^{-2} \div 3 \]

### Step 4: Combine the powers of 2.
Using the property \( a^m \times a^n = a^{m+n} \):
\[ 2^2 \times 2^{-2} = 2^{2 - 2} = 2^0 = 1 \]

### Step 5: Substitute back into the expression.
Now we have:
\[ 1 \div 3 \]

### Step 6: Simplify the division.
The final result is:
\[ \frac{1}{3} \]

Thus, the answer to the expression \( 4 \times (256)^{-\frac{1}{4}} \div (243)^{\frac{1}{5}} \) is \( \frac{1}{3} \).

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