Simplify
8.3^1/2 /( 6^1/2 + 5^1/2)
To simplify the given expression, 8.3^(1/2) / (6^(1/2) + 5^(1/2)), we can follow these steps:
Step 1: Simplify the square roots.
The square root of 8.3 can be written as (8.3)^(1/2) = √8.3, and the square root of 6 can be written as (6)^(1/2) = √6. Similarly, the square root of 5 can be written as (5)^(1/2) = √5.
So, the expression becomes: √8.3 / (√6 + √5).
Step 2: Rationalize the denominator.
To rationalize the denominator, we need to eliminate the square roots from the denominator. We can achieve this by multiplying both the numerator and denominator by the conjugate of the denominator, which is √6 - √5.
Multiplying the numerator by the conjugate, we get: √8.3 * (√6 - √5).
Multiplying the denominator by the conjugate, we get: (√6 + √5) * (√6 - √5).
Step 3: Simplify the numerator and denominator.
In the numerator, (√8.3 * √6) - (√8.3 * √5) can be simplified to √(8.3 * 6) - √(8.3 * 5).
In the denominator, (√6 * √6) - (√5 * √5) can be simplified to (√6)^2 - (√5)^2, which becomes 6 - 5.
So, the expression is simplified to: (√(8.3 * 6) - √(8.3 * 5)) / (6 - 5).
Step 4: Further simplify the expression.
Calculating the values inside the square roots, we have: (√49.8 - √41.5) / 1.
Simplifying the square roots, we get: (√(49 * 1.8) - √(41 * 1.5)) / 1.
Calculating the square roots, we have: (√(49) * √(1.8) - √(41) * √(1.5)) / 1.
Since the square root of 49 is 7, and the square root of 1.8 and 1.5 cannot be simplified further, the expression becomes: (7 * √1.8 - √41 * √1.5) / 1.
Finally, we have the simplified expression: 7√1.8 - √41.