Let `a,bepsilonR^+` for which `60^a=3` and `60^b=5` then `12^((1-a-b)/(2(1-b)))` is equal to

Explanation

To solve the problem, we start with the given equations:

1. \( 60^a = 3 \)
2. \( 60^b = 5 \)

We need to find the value of \( 12^{\frac{1-a-b}{2(1-b)}} \).

### Step 1: Express \( a \) and \( b \) in terms of logarithms

Taking logarithm on both sides of the first equation:

\[ a \log 60 = \log 3 \implies a = \frac{\log 3}{\log 60} \]

Similarly, for the second equation:

\[ b \log 60 = \log 5 \implies b = \frac{\log 5}{\log 60} \]

### Step 2: Substitute \( a \) and \( b \) into the expression

Now we substitute \( a \) and \( b \) into the expression \( 12^{\frac{1-a-b}{2(1-b)}} \):

\[ 1 - a - b = 1 - \frac{\log 3}{\log 60} - \frac{\log 5}{\log 60} \]

Combining the terms:

\[ 1 - a - b = 1 - \frac{\log 3 + \log 5}{\log 60} = 1 - \frac{\log(3 \times 5)}{\log 60} = 1 - \frac{\log 15}{\log 60} \]

Now, we can express \( 1 \) in terms of logarithms:

\[ 1 = \frac{\log 60}{\log 60} \]

Thus,

\[ 1 - a - b = \frac{\log 60 - \log 15}{\log 60} = \frac{\log \left(\frac{60}{15}\right)}{\log 60} = \frac{\log 4}{\log 60} \]

### Step 3: Calculate \( 2(1-b) \)

Next, we calculate \( 1 - b \):

\[ 1 - b = 1 - \frac{\log 5}{\log 60} = \frac{\log 60 - \log 5}{\log 60} = \frac{\log \left(\frac{60}{5}\right)}{\log 60} = \frac{\log 12}{\log 60} \]

Thus,

\[ 2(1-b) = 2 \cdot \frac{\log 12}{\log 60} = \frac{2 \log 12}{\log 60} \]

### Step 4: Substitute back into the exponent

Now substituting back into the exponent:

\[ \frac{1-a-b}{2(1-b)} = \frac{\frac{\log 4}{\log 60}}{\frac{2 \log 12}{\log 60}} = \frac{\log 4}{2 \log 12} \]

### Step 5: Simplify the expression

This simplifies to:

\[ \frac{\log 4}{2 \log 12} = \frac{\log 4}{\log 12^2} = \log_{12^2} 4 = \log_{144} 4 \]

### Step 6: Find \( 12^{\frac{1-a-b}{2(1-b)}} \)

Now we can find:

\[ 12^{\frac{1-a-b}{2(1-b)}} = 12^{\log_{144} 4} \]

Using the property \( a^{\log_b c} = c^{\log_b a} \):

\[ 12^{\log_{144} 4} = 4^{\log_{144} 12} \]

Since \( 12 = 4^{\frac{3}{2}} \):

\[ = 4^{\frac{3}{2} \cdot \log_{144} 4} = 4^{\frac{3}{2} \cdot \frac{1}{2}} = 4^{\frac{3}{4}} = 2^{3/2} = 2\sqrt{2} \]

Thus, the final answer is:

\[ \boxed{2} \]