Question

HELP ASAP, PLEASE!!! 100 POINTS!!!

Radical Expressions

A): Explain the error in this simplification.
B): Show your work as you correct the error.

Answer 1

Answer:

$\displaystyle \sqrt[14]{x^3}$

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

$\sqrt[5]{x^3}$

is equivalent to

$\displaystyle x^{\frac{3}{5}}$

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

$\displaystyle \frac{\sqrt[7]{x^5}}{\sqrt[4]{x^2}}=\frac{x^{\frac{5}{7}}}{x^{\frac{2}{4}}}$

$\displaystyle = x^{\frac{5}{7}-\frac{2}{4} }$

$\displaystyle = x^{\frac{3}{14} }$

$\displaystyle \boxed{ \sqrt[14]{x^3}}$