`(1^(3)+2^(3)+3^(3)+4^(3))^(-3//2)=`

Options

A. `10^(-3)`
B. `10^(-2)`
C. `10^(-4)`
D. `10^(-1)`

Answer

To solve the expression \((1^{3}+2^{3}+3^{3}+4^{3})^{(-3/2)}\), we will break it down step by step.

### Step 1: Calculate the cubes
First, we need to calculate the cubes of the numbers from 1 to 4.

\[ 1^3 = 1 \]
\[ 2^3 = 8 \]
\[ 3^3 = 27 \]
\[ 4^3 = 64 \]

### Step 2: Sum the cubes
Next, we sum these values together.

\[ 1 + 8 + 27 + 64 = 100 \]

### Step 3: Raise the sum to the power of \(-3/2\)
Now, we take the result from Step 2 and raise it to the power of \(-3/2\).

\[ (100)^{-3/2} \]

### Step 4: Simplify the exponent
The exponent \(-3/2\) can be interpreted as follows:

\[ (100)^{-3/2} = \frac{1}{(100)^{3/2}} \]

### Step 5: Calculate \(100^{3/2}\)
To compute \(100^{3/2}\), we can first find \(100^{1/2}\) (which is the square root of 100) and then raise it to the power of 3.

\[ 100^{1/2} = 10 \]
\[ 10^3 = 1000 \]

### Step 6: Substitute back
Now we substitute back into our expression:

\[ (100)^{-3/2} = \frac{1}{1000} \]

### Final Answer
Thus, the final result is:

\[ \frac{1}{1000} \]

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