If `a^(6)b^(3)`=4816 and `(a^10)/(b)`=301, what is the value of `(a^(2))/(b^(2))` ?

Options

A. 301
B. 64
C. 16
D. `1/4`

Answer

To solve the problem, we are given two equations:

1. \( a^6 b^3 = 4816 \)
2. \( \frac{a^{10}}{b} = 301 \)

We need to find the value of \( \frac{a^2}{b^2} \).

### Step 1: Rewrite the equations
We can rewrite the second equation as:
\[ a^{10} = 301b \]

### Step 2: Divide the two equations
Now, we can divide the second equation by the first equation:
\[ \frac{a^{10}/b}{a^6 b^3} = \frac{301}{4816} \]

### Step 3: Simplify the left side
Using the laws of exponents, we can simplify the left side:
\[ \frac{a^{10}}{b} \cdot \frac{1}{a^6 b^3} = \frac{a^{10 - 6}}{b^{1 + 3}} = \frac{a^4}{b^4} \]
Thus, we have:
\[ \frac{a^4}{b^4} = \frac{301}{4816} \]

### Step 4: Simplify the fraction
Now, we simplify \( \frac{301}{4816} \):
\[ \frac{301}{4816} = \frac{1}{16} \]
This can be verified by dividing both the numerator and the denominator by 301.

### Step 5: Take the square root
Now, we take the square root of both sides:
\[ \frac{a^2}{b^2} = \sqrt{\frac{1}{16}} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \frac{a^2}{b^2} \) is:
\[ \frac{1}{4} \]