Problem 66 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[6]{\frac{1}{64}}$$

Step-by-Step Solution

Answer

The result of evaluating the expression \(\sqrt[6]{\frac{1}{64}}\) is \(\frac{1}{2}\)

Step 1: Assess the root

First consider the number under the root, which is a fraction: \(\frac{1}{64}\). This is a real number and it can be evaluated.

Step 2: Rewrite as power of 6

Next, note that the denominator of our fraction, 64, can be expressed as \(2^6\). Therefore, our fraction, \(\frac{1}{64}\), can be written as \(\frac{1}{2^6}\) or \(2^{-6}\).

Step 3: Evaluate the expression

The formula for the sixth root of a number is \(n^{1/6}\). Therefore, we have \((2^{-6})^{1/6} = 2^{-6 * 1/6} = 2^{-1}\). This simplifies to \(\frac{1}{2}\).

Key Concepts

Evaluating Expressions

Evaluating expressions involves simplifying them to find their numerical value. The expression we have here is \( \sqrt[6]{\frac{1}{64}} \). The goal is to simplify this expression by finding the sixth root of \( \frac{1}{64} \). Evaluating involves identifying the numbers or variables involved, rewriting the expression in a simpler form, and computing the final value.

• Change complex numbers to rational numbers.
• Simplify the fractional parts when possible.
• Apply mathematical rules such as exponent rules and root evaluation.

In our example, 64 can be simplified to a power of 2, which helps in understanding how to evaluate the expression more easily. Breaking down the problem into smaller parts often makes the expression easier to manage.

Fractional Exponents

Fractional exponents are a way to express roots using powers. This is key in simplifying expressions like \( \sqrt[6]{\frac{1}{64}} \). A fractional exponent such as \( a^{m/n} \) means taking the \( n \)-th root of \( a \) and then raising it to the \( m \)-th power, or vice versa.

• The base, \( a \), is the number being operated on.
• The denominator of the fraction, \( n \), represents the root.
• The numerator, \( m \), represents the power.

In the example, rewriting \( \frac{1}{64} \) as \( 2^{-6} \) allows us to utilize the fractional exponent by expanding it to \( (2^{-6})^{1/6} = 2^{-6 \times 1/6} = 2^{-1} \). This simplifies the computation to get the result \( \frac{1}{2} \). Utilizing these principles can make calculating roots and powers more straightforward and intuitive.

Roots of Numbers

Roots of numbers are the opposite of powers, and they come in handy when you want to simplify expressions or reverse the effect of exponentiation. In this example, you are dealing with a sixth root: \( \sqrt[6]{a} \).

• The root indicates the number of times a number is multiplied by itself to reach \( a \).
• \( \sqrt[6]{x} \) refers to a number which, when raised to the power of 6, results in \( x \).

For \( \frac{1}{64} \), identifying 64 as \( 2^6 \) enables you to easily state that the sixth root is \( 2^{-1} \) or \( \frac{1}{2} \). Knowing exponent rules and the fundamental principle that roots can be represented as fractional exponents greatly aids in swiftly evaluating the expression correctly and revealing that it indeed results in a real number.