Problem 113 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ \sqrt[6]{64 a^{6} b^{6}} $$

Step-by-Step Solution

Answer

The simplified expression is \(2ab\).

Step 1: Break down the expression inside the radical

First, identify the components inside the radical: \(64\), \(a^6\), and \(b^6\). Notice that each of these terms has some power related to the sixth root we are dealing with.

Step 2: Simplify each component using the sixth root

Calculate the sixth root of each term separately. The sixth root of \(64\) is \(2\), because \(2^6 = 64\). For \(a^6\), the sixth root is \(a\), since \((a^1)^6 = a^6\). Similarly, of \(b^6\), the sixth root is \(b\), since \((b^1)^6 = b^6\).

Step 3: Combine the simplified components

Combine the results from the sixth root evaluation: \(2\), \(a\), and \(b\). So, \( \sqrt[6]{64 a^{6} b^{6}} = 2ab \).

Step 4: Final Solution

The simplified form of the original radical expression is \(2ab\).

Key Concepts

Understanding the Sixth Root

The sixth root is a special type of root in mathematics, similar to square roots or cube roots, but adjusted for six as the root index. When you encounter a sixth root in algebra like \( \sqrt[6]{x} \), it essentially asks for a number that, when multiplied by itself six times, equals \( x \).

Consider the expression \( \sqrt[6]{64} \). To find the sixth root, you are searching for a number \( y \) such that \( y^6 = 64 \). In this case, the answer is \( y = 2 \), because \( 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \).

This process is straightforward when the number inside the root is a perfect sixth power, meaning it can be expressed like \( y^6 \). This is a key idea in simplifying radical expressions.

Breaking Down Radical Expressions

A radical expression involves roots, and simplifying them often requires breaking them down into their basic components. For instance, in the expression \( \sqrt[6]{64 a^{6} b^{6}} \), the idea is to assess each term individually.

The first step is to identify each component inside the radical:

• \(64\) is a number which can be expressed in terms of its powers.
• \(a^6\) is a variable term raised to the sixth power.
• \(b^6\) is another variable term to the sixth power.

You treat each of these separately to simplify the whole radical. For each sixth power term like \(a^6\), find its sixth root, which is \(a\). This applies to \(b^6\) as well. Analyzing expressions this way allows you to clearly see which pieces have perfect root values and which need further simplification.

Steps in Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This makes solving equations and understanding them a lot easier. In our original problem, we're simplifying \( \sqrt[6]{64 a^6 b^6} \), which means finding a simpler equivalent without changing its value.

Here’s a step-by-step breakdown you can apply:

• Identify the sixth power terms inside the radical: 64, \(a^6\), and \(b^6\).
• Take the sixth root of each term individually.
• The sixth root of \(64\) is 2.
• Take the sixth root of \(a^6\), resulting in \(a\).
• Do the same for \(b^6\), resulting in \(b\).
• Combine these results: join them back together to get \(2ab\).

This simplification means the original expression \( \sqrt[6]{64 a^6 b^6} \) reduces cleanly to \(2ab\), making it easier to handle in further mathematical operations or calculations.