Problem 88 - Intermediate Algebra

Question

Evaluate each radical expression, if possible, without using a calculator. See Example 8. $$ \sqrt[6]{64} $$

Step-by-Step Solution

Answer

The sixth root of 64 is 2.

Step 1: Understand the Radical Expression

The given expression is \( \sqrt[6]{64} \). This means we need to find the number that, when raised to the 6th power, equals 64.

Step 2: Rewrite 64 as Powers of 2

The number 64 can be expressed as a power of 2, specifically \( 64 = 2^6 \). This will help in simplifying the expression.

Step 3: Apply Radical Property

Using the property \( \sqrt[n]{a^n} = a \), where \( n = 6 \) in this case, we can simplify \( \sqrt[6]{64} = \sqrt[6]{2^6} = 2 \). This means the sixth root of 64 is 2.

Step 4: Verify the Solution

To ensure our solution is correct, we verify by computing \( 2^6 \). Calculating, \( 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \), which matches the original radicand.

Key Concepts

Sixth Roots

In mathematics, a sixth root of a number is a value that, when raised to the sixth power, gives the original number. This is denoted as \( \sqrt[6]{x} \). In simpler terms, if you multiply the sixth root of a number by itself six times, you end up with the original number. In our example, we were tasked with finding \( \sqrt[6]{64} \), which involves determining a number that satisfies the equation \( x^6 = 64 \). To understand sixth roots further, remember:

• The "index" of the radical tells you which root you're dealing with—in this case, 6.
• This is different from square roots (index of 2) or cube roots (index of 3).
• Finding a sixth root relies on knowing the number can be expressed as a power—like how 64 is \( 2^6 \).

By identifying that 64 is a perfect sixth power (\( 64 = 2^6 \)), we find that the sixth root is simply 2.

Simplifying Radical Expressions

Simplifying radical expressions involves breaking down the expression into its most basic, controller components. For example, \( \sqrt[6]{64} \) was simplified by expressing 64 as \( 2^6 \). This transformation is possible because 64 is a power of a smaller base number, specifically 2.Consider the basic steps to simplify:

• Express the number under the radical as a power of its prime factors.
• Apply the radical rule \( \sqrt[n]{a^n} = a \).
• Where necessary, ensure any remaining exponents are less than the index.

In this scenario, the simplification works seamlessly because the exponent in \( 64 = 2^6 \) matches the index 6 in \( \sqrt[6]{64} \), allowing for direct simplification to 2. This process is essential in making complex expressions more manageable.

Properties of Exponents

The properties of exponents are crucial in simplifying radical expressions. These properties provide guidelines for handling different exponent operations in any expression. Let's review a few key ones:

• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Product: \((ab)^n = a^n \times b^n\)

In our example with \( 64 = 2^6 \), we used the properties of exponents to express 64 in a form that aligned with the radicand's index 6. By knowing that a number raised to its corresponding root's power simplifies directly—here \( \sqrt[6]{2^6} = 2 \)—we employed the power of a power property such that the expression inside the radical simplifies elegantly. These transformation techniques are integral for converting complex expressions into something simpler.