Respuesta :
Let us solve our both expressions one by one.
1. [tex]\frac{x^{\frac{1}{2}}} {x^{\frac{1}{4}}}[/tex]
To solve our first problem we will use quotient rule of exponents.
Let us subtract [tex]\frac{1}{4}[/tex] from [tex]\frac{1}{2}[/tex].
[tex]x^{\frac{1}{2}-\frac{1}{4}}[/tex]
[tex]x^{\frac{2-1}{4}}[/tex]
[tex]x^{\frac{1}{4}}[/tex]
Therefore, our expression [tex]\frac{x^{\frac{1}{2}}} {x^{\frac{1}{4}}}[/tex] simplifies to [tex]x^{\frac{1}{4}}[/tex].
2. For our second expression, we will use product rule of exponents to multiply all the x term inside the radical.
[tex]\sqrt[12]{x^{3}\cdot x^{5}\cdot x}[/tex]
[tex]\sqrt[12]{x^{(3+5+1)}}[/tex]
[tex]\sqrt[12]{x^{(9)}}[/tex]
Now we will express our radical into exponent form using power rule of exponent.
[tex](x^{9}) ^{\frac{1}{12}[/tex]
[tex]x^{\frac{1\cdot 9}{12} = x^{\frac{9}{12}[/tex]
Upon reducing our fraction we will get,
[tex]x^{\frac{3}{4}[/tex]
We can express our expression in radical form as: [tex]\sqrt[4]{x^{3}}[/tex].
