We are asked to analyze the given radical equation.
sqrt(a)+sqrt(b)=sqrt(a+b)
While we have the Multiplication and Division Properties of Square Roots, there is no such thing as the Addition Property of Square Roots. Because of this, the sum of square roots is not always equal to the square root of the sum. To make sure, let's consider a few examples.
Sum of Square Roots
Square Root of the Sum
Are the Values the Same?
sqrt(2)+sqrt(2)≈ 2.83
sqrt(2+2)= 2
No
sqrt(4)+sqrt(25)= 7
sqrt(4+25)≈ 5.38
No
sqrt(5)+sqrt(11)≈ 5.55
sqrt(5+11)= 4
No
sqrt(7)+sqrt(13)≈ 6.25
sqrt(7+13)≈ 4.47
No
As we can see, in each of these examples, we obtained different values for each side of our equation. Therefore, we can disregard the possibility of the answer being always. Now, let's think about whether this equation can be true for some specific values. What if a= 0 and b= 0?
sqrt(0)+sqrt(0)&=sqrt(0+ 0)
&⇓
0&=0 ✓
When both a and b equal 0, the equation is true. However, if we try any other values, it will be false. This allows us to conclude that the given equation is sometimes true.
