Find the value of `{(49)^((3)/(2))+(49)^(-(3)/(2))}`

Options

A. `(117549)/(343)`
B. `(117550)/(343)`
C. `(117659)/(343)`
D. `(117650)/(343)`

Answer

To solve the expression \( (49)^{\frac{3}{2}} + (49)^{-\frac{3}{2}} \), we can follow these steps:

### Step 1: Rewrite the base
First, we can express 49 as a power of 7:
\[ 49 = 7^2 \]
Thus, we can rewrite the expression:
\[ (49)^{\frac{3}{2}} = (7^2)^{\frac{3}{2}} \]

### Step 2: Apply the power of a power rule
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify:
\[ (7^2)^{\frac{3}{2}} = 7^{2 \cdot \frac{3}{2}} = 7^3 \]
Now we can also rewrite the second term:
\[ (49)^{-\frac{3}{2}} = (7^2)^{-\frac{3}{2}} = 7^{2 \cdot -\frac{3}{2}} = 7^{-3} \]

### Step 3: Combine the terms
Now we have:
\[ 7^3 + 7^{-3} \]

### Step 4: Rewrite \(7^{-3}\)
We can rewrite \(7^{-3}\) as:
\[ 7^{-3} = \frac{1}{7^3} \]
Thus, the expression becomes:
\[ 7^3 + \frac{1}{7^3} \]

### Step 5: Find a common denominator
To combine these fractions, we can find a common denominator:
\[ 7^3 + \frac{1}{7^3} = \frac{7^3 \cdot 7^3 + 1}{7^3} = \frac{7^6 + 1}{7^3} \]

### Step 6: Calculate \(7^6 + 1\)
Now we need to calculate \(7^6\):
\[ 7^6 = (7^3)^2 = 343^2 \]
Calculating \(343^2\):
\[ 343^2 = 117649 \]
Thus:
\[ 7^6 + 1 = 117649 + 1 = 117650 \]

### Step 7: Final expression
Now we can write the final expression:
\[ \frac{117650}{343} \]

### Conclusion
The value of \( (49)^{\frac{3}{2}} + (49)^{-\frac{3}{2}} \) is:
\[ \frac{117650}{343} \]