Evaluate each expression in Exercises 55–66, or indicate that the root is not a real number. ⁶√1/64

Verified step by step guidance

1. Identify the expression: $\sqrt[6]{\frac{1}{64}}$.
2. Recognize that $\frac{1}{64}$ can be rewritten as ${64}^{-1}$.
3. Recall that $64=2^6$, so ${64}^{-1}=(2^6{)}^{-1}=2^{-6}$.
4. Apply the property of exponents: $\sqrt[6]{2^{-6}}=(2^{-6}{)}^{\frac{1}{6}}$.
5. Simplify the expression using the power of a power property: $(2^{-6}{)}^{\frac{1}{6}}=2^{-1}$.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. The notation '⁶√' indicates the sixth root of a number, which is the value that, when raised to the sixth power, equals the original number. Understanding how to manipulate and evaluate these expressions is crucial for solving problems involving roots.

Rational Exponents

Rational exponents provide an alternative way to express roots. For example, the sixth root of a number can be expressed as that number raised to the power of 1/6. This concept allows for easier manipulation of expressions, especially when combined with other algebraic operations.

Real vs. Complex Numbers

In algebra, it's important to distinguish between real and complex numbers. A real number is any value along the number line, while complex numbers include an imaginary unit 'i' (where i² = -1). When evaluating roots, if the expression under the root is negative, the result will be a complex number, indicating that the root is not a real number.