In Exercises 59–76, find the indicated root, or state that the expression is not a real number.__⁶√64

Verified step by step guidance

1. Recognize that the expression \( \sqrt[6]{64} \) is asking for the sixth root of 64.
2. Recall that finding the sixth root of a number is equivalent to raising that number to the power of \( \frac{1}{6} \).
3. Express 64 as a power of a smaller base. Note that 64 can be written as \( 2^6 \).
4. Apply the property of exponents: \( (a^m)^n = a^{m \cdot n} \). In this case, \( (2^6)^{\frac{1}{6}} = 2^{6 \cdot \frac{1}{6}} \).
5. Simplify the expression \( 2^{6 \cdot \frac{1}{6}} \) to find the sixth root of 64.

Key Concepts

Roots and Radicals

Roots and radicals are mathematical operations that involve finding a number that, when raised to a certain power, yields a given value. The notation for the nth root of a number 'a' is expressed as n√a, where 'n' indicates the degree of the root. Understanding how to simplify and compute roots is essential for solving problems involving radical expressions.

Even and Odd Roots

Even roots, such as square roots and fourth roots, can yield both positive and negative results, but they are typically defined to return only the principal (non-negative) root. In contrast, odd roots, like cube roots and fifth roots, can yield negative results as well. Recognizing the difference between even and odd roots is crucial for determining the nature of the solutions.

Real Numbers

Real numbers include all the rational and irrational numbers that can be found on the number line. When evaluating roots, it is important to determine whether the result is a real number. For example, the even root of a negative number is not a real number, while the odd root of a negative number is. This distinction is vital for correctly interpreting the results of root calculations.