Respuesta :
Having the following product:
[tex](\sqrt[]{5x})\cdot(\sqrt[]{x+3})[/tex]We need to establish the values of x for which it is defined.
The product has 2 square roots. We know that square roots are defined in real numbers only when its argument is 0 or a positive number.
We can begin checking the first one: √(5x). The argument of the square root (5x) has to be equal or larger than 0, then:
[tex]\begin{gathered} 5x\ge0 \\ \end{gathered}[/tex]We can divide both sides by 5, having:
[tex]x\ge0[/tex]The first factor is defined for values of x larger or equal to 0.
Following the same logic for the other factor:
[tex]\begin{gathered} x+3\ge0 \\ x\ge-3 \end{gathered}[/tex]Then, we have two restrictions so far:
[tex]\begin{gathered} x\ge0 \\ x\ge-3 \end{gathered}[/tex]The restrictions for the product of both radicals will be the intersection of the two previous conditions. The first restriction goes from 0 to infinity and the second one from -3 to infinity. The intersection of both intervals gives the numbers from 0 to infinity. Then, the values of x for which the product is defined are:
[tex]x\ge0[/tex]Correct option is D.
