NEED HELP ASAP What is the simplified form of the following expression? 4 sqrt 3/2x Assume x>0

Accepted Solution

Answer:[tex]\frac{4\sqrt{6x}}{2x}[/tex]Explanation:The problem we are given is[tex]4\sqrt{\frac{3}{2x}}[/tex]We can write the square root of a fraction as a fraction with a separate radical for the numerator and denominator; this gives us[tex]4\times \frac{\sqrt{3}}{\sqrt{2x}}[/tex]We can write the whole number 4 as the fraction 4/1; this gives us[tex]\frac{4}{1}\times \frac{\sqrt{3}}{\sqrt{2x}}\\\\=\frac{4\sqrt{3}}{\sqrt{2x}}[/tex]We now need to "rationalize the denominator."  This means we need to cancel the square root in the denominator.  In order to do this, we multiply both numerator and denominator by √(2x); this is because squaring a square root will cancel it:[tex]\frac{4\sqrt{3}}{\sqrt{2x}}\times \frac{\sqrt{2x}}{\sqrt{2x}}\\\\=\frac{4\sqrt{3}\times \sqrt{2x}}{2x}[/tex]When multiplying radicals, we can extend the radical over both factors:[tex]\frac{4\sqrt{3} \times \sqrt{2x}}{2x}\\\\=\frac{4\sqrt{3\times 2x}}{2x}\\\\=\frac{4\sqrt{6x}}{2x}[/tex]