Short Answer 
This response outlines the process of understanding and applying the rules of indices in mathematics, highlighting key multiplication and division rules. It also emphasizes the importance of accurate calculations, illustrating the correct simplification of an expression involving indices.
Step 1: Understanding Indices
In mathematics, indices are expressions that represent the number of times a number or variable is multiplied by itself. It’s essential to comprehend the rules governing these expressions, which include the multiplication and division laws. Key rules to remember are:
- Multiplication Rule: (a^m times a^n = a^{m+n})
- Division Rule: (frac{a^m}{a^n} = a^{m-n})
Step 2: Applying the Laws of Indices
When simplifying expressions involving indices, accurate application of the rules is crucial. Using the example ((frac{x^{frac{2}{5}} times x^{frac{4}{5}}}{x^{frac{2}{5}}})^{frac{1}{2}}), start by simplifying the numerator and denominator with the indices’ laws. Follow these steps:
- Combine the numerator using the multiplication rule: (x^{frac{2}{5} + frac{4}{5}} = x^{frac{6}{5}})
- Next, apply the division rule on the entire expression.
Step 3: Identifying Calculation Errors
It’s essential to double-check calculations to avoid mistakes. In the example, the student incorrectly assessed (6/5 – 2/5) as 3 instead of the correct value of (frac{4}{5}). This led to the wrong result. The correct iterations are:
- Evaluate (x^{frac{6}{5} – frac{2}{5}} = x^{frac{4}{5}})
- Finally, find the square root, resulting in (x^{frac{2}{5}}).