%5E1%2F2-simplify.jpeg "(4-x^2)^1/2 Simplify")
Simplifying the Expression (4-x^2)^1/2
The expression (4-x^2)^1/2 represents the square root of (4-x^2). While it's already in a simplified form, we can further manipulate it by recognizing the pattern of a difference of squares.
Understanding the Difference of Squares
The difference of squares pattern states that: a^2 - b^2 = (a + b)(a - b)
In our expression, we can rewrite 4 as 2^2. This allows us to apply the pattern:
(4 - x^2)^1/2 = (2^2 - x^2)^1/2
Now, we can apply the difference of squares formula:
(2^2 - x^2)^1/2 = [(2 + x)(2 - x)]^1/2
Simplifying Further
While this expression is fully factored, we can simplify it further by considering the square root of a product:
√(a * b) = √a * √b
Applying this to our expression:
[(2 + x)(2 - x)]^1/2 = √(2 + x) * √(2 - x)
Therefore, the fully simplified form of (4-x^2)^1/2 is √(2 + x) * √(2 - x).
Conclusion
By recognizing the pattern of a difference of squares and applying the properties of square roots, we were able to simplify the expression (4-x^2)^1/2 into a more readable and understandable form. This simplification can be useful for further calculations or analysis.