Simplify: $(-\sqrt{x-4})^2$
Answer (2)
The expression ( − x − 4 ) 2 simplifies to x − 4 . This is because squaring the negative sign yields 1, and squaring the square root returns the value inside the square root. Therefore, the final simplified result is x − 4 .
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The problem asks to simplify ( − x − 4 ) 2 .
Squaring the negative sign gives ( − 1 ) 2 = 1 .
Squaring the square root gives ( x − 4 ) 2 = x − 4 .
Combining these, the simplified expression is x − 4 .
Explanation
Understanding the Problem We are asked to simplify the expression ( − x − 4 ) 2 . Let's break this down step by step to make sure we understand each part. First, notice that the expression inside the square root, x − 4 , must be greater than or equal to zero, since we can only take the square root of non-negative numbers. This means that x ≥ 4 .
Simplifying the Expression Now, let's simplify the expression. We have ( − x − 4 ) 2 . This means we are squaring both the negative sign and the square root. When we square a negative sign, we get a positive sign, since ( − 1 ) 2 = 1 . When we square a square root, we get the expression inside the square root, so ( x − 4 ) 2 = x − 4 .
Combining the Results Putting it all together, we have ( − x − 4 ) 2 = ( − 1 ) 2 ( x − 4 ) 2 = 1 ⋅ ( x − 4 ) = x − 4 . So the simplified expression is x − 4 .
Final Answer Therefore, the simplified expression is x − 4 .
Examples
Imagine you are designing a garden and need to calculate the area of a square plot. If the side length of the square is given by x − 4 , then the area of the square is ( x − 4 ) 2 = x − 4 . Simplifying expressions like this helps you determine the actual area based on the value of x . This type of simplification is also useful in physics, where you might need to calculate kinetic energy or potential energy based on square roots of variables.
