Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁵√(−3)⁵

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Recognize that the given expression involves a fifth root: ⁵√((-3)^5). This means we are looking for the number that, when raised to the fifth power, equals (-3)^5.

Simplify the expression inside the root. The base is -3, and it is raised to the power of 5. Using the rule of exponents, calculate (-3)^5. Note that raising a negative number to an odd power results in a negative number.

Now, the expression becomes ⁵√(-243), since (-3)^5 equals -243.

Determine whether the fifth root of -243 is a real number. Recall that the nth root of a negative number is real if n is odd. Since 5 is odd, the fifth root of -243 is a real number.

Conclude that the fifth root of -243 is the number that, when raised to the fifth power, equals -243. This number is -3, as (-3)^5 = -243.

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