a student simplified the rational expression using the steps shown.\n left(\frac{x^{\frac{2}{5}}cdot…

a student simplified the rational expression using the steps shown.\n left(\frac{x^{\frac{2}{5}}cdot x^{\frac{4}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}=left(\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}=(x^{3})^{\frac{1}{2}} = x^{\frac{3}{2}} \nis the answer correct? explain.

a student simplified the rational expression using the steps shown.\n left(\frac{x^{\frac{2}{5}}cdot x^{\frac{4}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}=left(\frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}}\right)^{\frac{1}{2}}=(x^{3})^{\frac{1}{2}} = x^{\frac{3}{2}} \nis the answer correct? explain.

Answer

Explanation:

Step1: Check numerator simplification

When multiplying ( x^{\frac{2}{5}} \cdot x^{\frac{4}{5}} ), add exponents:
( \frac{2}{5} + \frac{4}{5} = \frac{6}{5} ), correct.

Step2: Analyze fraction simplification

The expression becomes ( \frac{x^{\frac{6}{5}}}{x^{\frac{2}{5}}} ).
Correct rule: Subtract exponents when dividing:
( \frac{6}{5} - \frac{2}{5} = \frac{4}{5} ), not ( 3 ).
Student incorrectly divided ( \frac{6}{5} \div \frac{2}{5} = 3 ) (wrong operation).

Step3: Apply outer exponent

Correctly, after step2, it should be ( \left( x^{\frac{4}{5}} \right)^{\frac{1}{2}} = x^{\frac{4}{5} \cdot \frac{1}{2}} = x^{\frac{2}{5}} ), not ( x^{\frac{3}{2}} ).

Answer:

No, the answer is incorrect. The student misapplied exponent rules by dividing instead of subtracting exponents when simplifying the fraction, leading to an erroneous result. The correct simplified form is ( x^{\frac{2}{5}} ).