(3^-2+4^-2)^-1/2
To simplify the expression (3^-2 + 4^-2)^(-1/2), we'll follow the order of operations (also known as PEMDAS or BODMAS):
Step 1: Evaluate the exponents within the parentheses:
3^-2 = 1/(3^2) = 1/9
4^-2 = 1/(4^2) = 1/16
So, the expression now becomes (1/9 + 1/16)^(-1/2).
Step 2: Combine the fractions with a common denominator:
To add fractions with different denominators, we need to find a common denominator. In this case, the least common denominator (LCD) is 144. We can now express 1/9 and 1/16 with the denominator of 144:
1/9 = (1/9) * (16/16) = 16/144
1/16 = (1/16) * (9/9) = 9/144
So, the expression becomes (16/144 + 9/144)^(-1/2).
Step 3: Add the fractions:
16/144 + 9/144 = 25/144
The expression now becomes (25/144)^(-1/2).
Step 4: Take the square root of the numerator and denominator:
To calculate the square root of a fraction, we take the square root of the numerator over the square root of the denominator:
√(25/144) = √25 / √144 = 5/12
The final simplified expression is 5/12.