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

To simplify the expression (3^-2 + 4^-2) ^ -1/2, we can follow these steps:

Step 1: Simplify the exponents
Simplify the exponents separately in the parentheses. The exponent -2 means that the base is being raised to the power of -2.

3^-2 = 1/3^2 = 1/9

4^-2 = 1/4^2 = 1/16

So, the expression simplifies to (1/9 + 1/16) ^ -1/2.

Step 2: Combine the fractions
Now simplify the fractions by finding a common denominator and adding them.
The common denominator of 9 and 16 is 144.

1/9 = (16/16)*(1/9) = 16/144

1/16 = (9/9)*(1/16) = 9/144

Now we have (16/144 + 9/144) ^ -1/2.

Step 3: Add the fractions
Add the numerators from the previous step:
16/144 + 9/144 = (16 + 9)/144 = 25/144

Now we have (25/144) ^ -1/2.

Step 4: Apply the exponent rule
To calculate the negative exponent, we need to invert the fraction and apply the positive exponent.
(25/144) ^ -1/2 = (144/25) ^ 1/2

Now we have (144/25) ^ 1/2.

Step 5: Evaluate the square root
To evaluate the square root, we can find the square root of the numerator and the square root of the denominator separately.

Square root of 144 = 12
Square root of 25 = 5

So, (144/25) ^ 1/2 becomes (12/5).

Final Answer: Therefore, the expression (3^-2 + 4^-2) ^ -1/2 simplifies to (12/5).