Question

Simplify the following DIVISION of radicals expression. Make sure you rationalize the denominator if needed. root (3x^3y)/(6xy^3)

a. x root (2)/y
b. x/2y
c. x root (2)/2y
d. None of the above

Answer 1

\[\sqrt{\frac{3x^3y}{6xy^3}} =\sqrt{\frac{3*x^3*y } {6*x*y^3 }} = \sqrt{\frac{3}{6}*\frac{x^3}{x}*\frac{y}{y^3}}\]
Simplify each fraction, then multiply the results together. At that point, recall that \[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]You rationalize if you end up with a radical sign in the denominator by multiplying both numerator and denominator by the denominator. In other words,
\[\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}*\sqrt{b}}{\sqrt{b}*\sqrt{b}} = \frac{\sqrt{ab}}b\]

Strictly speaking, to properly simplify this equation, we need to know whether \(x\) and \(y\) are both \(> 0\) because they appear as powers under the radical sign. For example:\[\sqrt{x^2} = x\text{ if }x>0\text{ but not otherwise}\]\[x=3:\]\[\sqrt{3*3} = 3\checkmark\]But look at what happens if \(x < 0\):\[\sqrt{(-3)*(-3)} = \sqrt{9} = 3\]Oops! \[\sqrt{x^2} \ne x,\,x<0\]