When is the following a true statement?
�ã(a+b) = �ãa + �ãb
if a and b are nonnegative.
To determine when the following statement is true:
√(a+b) = √a + √b, assuming a and b are nonnegative, we need to consider the properties of square roots.
One of the properties of square roots states that if we have two nonnegative real numbers, say x and y, then √(x+y) = √x + √y if and only if x and y are nonnegative and they have the same sign (both positive or both zero).
Now, let's apply this property to the given equation:
√(a+b) = √a + √b
For this equation to hold true, both sides must be equal, and we need to confirm that a and b are nonnegative and have the same sign.
First, since we are given that a and b are nonnegative, we have confirmed the first condition.
Next, we need to check if a and b have the same sign. To do this, we can compare the values of a and b to determine the signs.
1. If a and b are both positive, then the equation will hold true, as in this case, √a and √b will also be positive and their sum will be the positive square root of (a+b).
2. If a and b are both zero (0), then the equation will hold true because both sides will be equal to zero.
Therefore, the given statement √(a+b) = √a + √b is true if a and b are nonnegative and have the same sign (both positive or both zero).