Question

I would deeply appreciate it if you could help me with my assignment

Answer 1

Given the expression:

`√(4x)\cdot√(x+2)`

You need to apply the Property for Radicals that states that the Multiplication of two roots with the same indices is equal to the root of the Product. Then:

`=√((4x)(x+2))`

Apply the Distributive Property inside the square root:

`\begin{gathered} =√((4x)(x)+(4x)(2)) \\ \\ =√(4x^2+8x) \end{gathered}`

By definition, the Radicand (the value inside the square root) must be greater than or equal to zero:

`4x^2+8x\ge0`

Solving for "x", you get:

- Case 1

`\begin{gathered} x\ge0 \\ \\ x+2\ge0\Rightarrow x\ge-2 \end{gathered}`

- Case 2:

`\begin{gathered} x\leq0 \\ \\ x+2\leq0\Rightarrow x\leq-2 \end{gathered}`

By determining the Intersection, you get that the solution is:

`\begin{gathered} x\ge0 \\ x\leq-2 \end{gathered}`

Hence, the answer is: Option B and Option D.