In a simplification problem involving rational exponents and negative bases, evaluate the real value of the expression (−7776)^(2/5).

Introduction / Context:
This question checks your understanding of rational exponents and how to deal with powers of negative numbers. The expression (−7776)^(2/5) involves taking a fifth root and then squaring. It is important to recognise the base as a perfect fifth power, which allows the simplification to be carried out without decimals. This skill is useful in algebra, number theory, and many aptitude exams where time saving manipulation is important.

Given Data / Assumptions:

• The base is −7776, a negative integer.
• The exponent is 2/5, which can be interpreted as a fifth root followed by a square.
• We are interested in the real value of the expression.
• Standard rules for rational exponents apply: a^(m/n) = (n-th root of a)^m, where the root is taken in the real number system when possible.

Concept / Approach:
To evaluate (−7776)^(2/5), we can write the exponent as 2/5 = 2 * (1/5), so the expression becomes [ (−7776)^(1/5) ]^2. The fifth root of −7776 is a real number because the index 5 is odd. Recognising 7776 as a power of 6 greatly simplifies the work. Once the fifth root is found, we simply square the result to complete the calculation.

Step-by-Step Solution:
Step 1: Express the power as a composition: (−7776)^(2/5) = [ (−7776)^(1/5) ]^2.Step 2: Notice that 6^5 = 6 × 6 × 6 × 6 × 6 = 7776.Step 3: Therefore the real fifth root of 7776 is 6, and the real fifth root of −7776 is −6, since (−6)^5 = −7776.Step 4: So (−7776)^(1/5) = −6 in the real number system.Step 5: Now square this value: [ (−7776)^(1/5) ]^2 = (−6)^2 = 36.Step 6: Hence (−7776)^(2/5) = 36.

Verification / Alternative check:
We can verify by reversing the process. Take the value 36 and raise it to the power 5/2. Using exponent rules, 36^(5/2) = (√36)^5 = 6^5 = 7776. Since the original base was −7776, and we squared a negative fifth root, the sign information is lost at the squaring step, which is expected. For the real value of (−7776)^(2/5), we are only concerned with the non negative result of the even power, which is 36.

Why Other Options Are Wrong:
The numbers 16, 21, 64, and 6 do not satisfy the relation 36^(5/2) = 7776. For example, 16^(5/2) = (√16)^5 = 4^5 = 1024, which is much smaller. Similarly, 64^(5/2) = 8^5 = 32768, which is larger than 7776. The value 6 is only the fifth root in magnitude, not the final result after squaring. Thus, 36 is the only value that correctly matches the expression.

Common Pitfalls:
Students sometimes misinterpret a^(m/n) as a^m / a^n or try to take the square first and then the root without checking whether it is valid for negative bases. Another common error is to ignore the fact that the fifth root of a negative number is negative, leading to sign confusion. Thinking in terms of first finding the appropriate root and then applying the remaining power helps keep the logic clear and avoids such mistakes.

Final Answer:
The real value of the expression (−7776)^(2/5) is 36.