91Ó°ÊÓ

The concept of a fifth root involves finding a number that, when raised to the power of five, gives the original number. For example, the fifth root of 32 is 2, because \(2^5 = 32\). In our exercise, the given expression is \( \sqrt[5]{a^{5}} \). To simplify this, we utilize the property of radicals and exponents. When the index of the root (in this case, 5) matches the exponent, the expression simplifies to just the base. Hence, \( \sqrt[5]{a^5} \) simplifies to \( a \). This process is straightforward because 5 is an odd number and no additional steps involving absolute value are needed.

An exponent is called 'odd' if it is an odd number such as 1, 3, 5, etc. In our exercise, we are working with the fifth power, which is an odd exponent. When dealing with odd exponents, we have a special property: The odd root of a number raised to an odd exponent yields the base number itself. For example:

• \( \sqrt[5]{a^5} = a \)
• \( \sqrt[3]{b^3} = b \)

In both cases, since the root and the exponent match and are odd, the radicals simplify neatly without needing absolute values. This rule makes handling odd exponents quite convenient.

The concept of absolute value ensures that the result of a root operation stays non-negative. This is crucial when dealing with even roots, such as square roots or fourth roots, because the root of a negative number is not real. However, for odd roots (like cube roots or fifth roots), the concept of absolute value isn't necessary. This is because the odd roots of negative numbers are still real. In our exercise, the fifth root of \(a^5\), \( \sqrt[5]{a^5} \), simplifies directly to \( a \) with no need for absolute value notation. This is due to the property that odd roots can handle negative numbers without issues. Therefore, simplifying odd roots is typically more straightforward.