Answer:
[tex]\displaystyle \sqrt[14]{x^3}[/tex]
Step-by-step explanation:
Radical As A Fractional Exponent
We can write a radical as a fractional exponent. The power to which the base is raised is the numerator and the root is the denominator. For example, the radical
[tex]\sqrt[5]{x^3}[/tex]
is equivalent to
[tex]\displaystyle x^{\frac{3}{5}}[/tex]
A) The simplification shown in the image is wrong because the student subtracted the roots of the radicals separated from the subtraction of the powers.
B) The correct procedure is
* Express both radicals as fractional exponents
* Subtract both exponents
* Simplify the resultant fraction
* Return the fractional exponent to radical form
In our case, the correct procedure is
[tex]\displaystyle \frac{\sqrt[7]{x^5}}{\sqrt[4]{x^2}}=\frac{x^{\frac{5}{7}}}{x^{\frac{2}{4}}}[/tex]
[tex]\displaystyle = x^{\frac{5}{7}-\frac{2}{4} }[/tex]
[tex]\displaystyle = x^{\frac{3}{14} }[/tex]
[tex]\displaystyle \boxed{ \sqrt[14]{x^3}}[/tex]
