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

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

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^{(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^{(5)/(7)}}{x^{(2)/(4)}}`

`\displaystyle = x^{(5)/(7)-(2)/(4) }`

`\displaystyle = x^{(3)/(14) }`

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