`[169^(-3)/((196)^(-8))]^(1/48)=`

Options

A. `14^(1/3)/13^(1/8)`
B. `14^((-1)/3)/13^((-1)/8)`
C. `13^(1/8)/14^(1/3)`
D. `14^(1/8)/13^(1/3)`

Explanation

To solve the expression \([169^{-3}/(196)^{-8}]^{1/48}\), we can follow these steps:

### Step 1: Rewrite the bases in terms of their prime factors
We know that:
- \(169 = 13^2\)
- \(196 = 14^2\)

So we can rewrite the expression:
\[ 169^{-3} = (13^2)^{-3} = 13^{-6} \]
\[ 196^{-8} = (14^2)^{-8} = 14^{-16} \]

### Step 2: Substitute the rewritten bases into the expression
Now substitute these back into the original expression:
\[ \left[\frac{13^{-6}}{14^{-16}}\right]^{1/48} \]

### Step 3: Apply the property of negative exponents
Using the property \(\frac{a^{-m}}{b^{-n}} = \frac{b^n}{a^m}\), we can rewrite the fraction:
\[ \frac{13^{-6}}{14^{-16}} = 13^{-6} \cdot 14^{16} \]
Thus, the expression becomes:
\[ \left[14^{16} \cdot 13^{-6}\right]^{1/48} \]

### Step 4: Apply the exponent to both the numerator and denominator
Using the property \((a \cdot b)^m = a^m \cdot b^m\):
\[ = \left(14^{16}\right)^{1/48} \cdot \left(13^{-6}\right)^{1/48} \]
\[ = 14^{\frac{16}{48}} \cdot 13^{\frac{-6}{48}} \]

### Step 5: Simplify the exponents
Now simplify the fractions:
\[ = 14^{\frac{1}{3}} \cdot 13^{-\frac{1}{8}} \]

### Step 6: Rewrite the expression
This can be rewritten as:
\[ = \frac{14^{\frac{1}{3}}}{13^{\frac{1}{8}}} \]

### Final Answer
Thus, the final simplified expression is:
\[ \frac{14^{\frac{1}{3}}}{13^{\frac{1}{8}}} \]