We need to simplify [tex] \frac{ \sqrt{14x^3} }{ \sqrt{18x} } [/tex]
First lets factor [tex] \sqrt{14x^3} [/tex]
[tex] \sqrt{14x^3} [/tex] = [tex] \sqrt{14} \sqrt{x^3}[/tex]
[tex] \sqrt{14} = \sqrt{2} \sqrt{7} [/tex] by applying the radical rule [tex] \sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b} [/tex]
[tex] \sqrt{x^3} = x^{3/2}[/tex] By applying the radical rule [tex] \sqrt[n]{x^m} = x^{m/n}[/tex]
So
[tex] \sqrt{14x^3} [/tex] = [tex] \sqrt{14} \sqrt{x^3}[/tex] = [tex] \sqrt{2} \sqrt{7}x^{3/2} [/tex]
Now let's factor [tex] \sqrt{18x} [/tex]
By applying the radical rule [tex] \sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b} [/tex],
[tex]\sqrt{18x} = \sqrt{18} \sqrt{x} [/tex]
[tex] \sqrt{18} = \sqrt{2} * 3[/tex]
So [tex] \sqrt{18x} [/tex] = [tex] \sqrt{2}*3 \sqrt{x} [/tex]
So [tex] \frac{ \sqrt{14x^3} }{ \sqrt{18x} } [/tex] = [tex] \frac{ \sqrt{2} \sqrt{7} x^{3/2} }{ \sqrt{2}*3 \sqrt{x} } [/tex]
We know that [tex] \sqrt[n]{x} = x^{1/n}[/tex] so [tex] \sqrt{x} = x^{1/2}[/tex]
We now have [tex]\frac{ \sqrt{2} \sqrt{7} x^{3/2} }{ \sqrt{2}*3 \sqrt{x}} = \frac{ \sqrt{2} \sqrt{7} x^{3/2} }{ \sqrt{2}*3x^{1/2}} [/tex]
We know that [tex] \frac{x^a}{x^b} = x^{a-b}[/tex]
So [tex] \frac{x^{3/2}}{x^{1/2}} = x^{3/2 - 1/2} = x[/tex]
We now got [tex]\frac{ \sqrt{2} \sqrt{7} x^{3/2} }{ \sqrt{2}*3x^{1/2}} = \frac{ \sqrt{2} \sqrt{7} x }{ \sqrt{2}*3}
[/tex]
We can notice that the numerator and the denominator both got √2 in a multiplication, so we can simplify them, and we get:
[tex]\frac{ \sqrt{2} \sqrt{7} x }{ \sqrt{2}*3} = \frac{ \sqrt{7}x }{3} [/tex]
All in All, we get [tex] \frac{ \sqrt{14x^3} }{ \sqrt{18x} } [/tex] = [tex] \frac{ \sqrt{7}x }{3} [/tex]
Hope this helps! :D
