Problem 57 - Precalculus Mathematics for Calculus

Question

Simplify the expression. Assume that the letters denote any real numbers. $$\sqrt[6]{64 a^{6} b^{7}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 2ab \sqrt[6]{b} \).

Step 1: Recognize the Radical Expression

The given expression is \( \sqrt[6]{64 a^{6} b^{7}} \), which is a sixth root of the entire expression inside the radical sign.

Step 2: Simplify the Constant

Let's consider the constant 64, which can be rewritten as \( 2^6 \) because \( 2^6 = 64 \). This allows us to extract it from the sixth root as 2.

Step 3: Simplify the Variable Part for \( a \)

For the term \( a^6 \) under the sixth root, we can take \( a \) out of the root because the power is equal to the root: \( \sqrt[6]{a^6} = a \).

Step 4: Simplify the Variable Part for \( b \)

The term \( b^7 \) can be separated into \( b^6 \cdot b^1 \). This allows us to take \( b \) out of the sixth root since \( \sqrt[6]{b^6} = b \), leaving \( \sqrt[6]{b} \) inside.

Step 5: Combine and Simplify

Combine all parts extracted and those left under the root. We have 2, \( a \), and \( b \) taken out with \( \sqrt[6]{b} \) remaining inside the root. The expression simplifies to \( 2ab \sqrt[6]{b} \).

Key Concepts

Sixth Root Simplification

Simplifying a sixth root might sound complicated, but it's quite manageable by understanding the process of breaking down the terms inside the root. The sixth root is like asking for a number that, when multiplied by itself six times, gives us another number. For example, for the sixth root of 64, we need a number which, raised to the power of 6, equals 64. Simplification involves:

• Identifying perfect sixth powers inside the radical, such as numbers like 64, which can be broken down to simpler forms.
• Recognizing terms like \( a^6 \), that are perfect sixth powers, meaning they can be pulled out of the radical as their base, \( a \).

Once you clearly identify these components, removing them from the radical process becomes straightforward. This simplifies the entire expression and makes it easier to handle.

Exponent Properties

Exponents play a crucial role in simplifying radical expressions. Understanding exponent rules is essential for making complex expressions simpler. Here's how they help:

• The rule \( x^{n/m} = \sqrt[m]{x^n} \) can be especially helpful. This translates an exponent into a radical, showing the relationship between roots and powers.
• Exponent properties also guide us in breaking up powers, such as splitting \( b^7 \) into \( b^6 \times b^1 \). This allows us to handle the expression in more manageable parts.

By applying these properties correctly, you'll find radical expressions becoming less intimidating and more structured, leading to a simplified form.

Radical Expressions

Radical expressions are any expressions that include roots. They can often seem challenging at first glance, but breaking them down into simpler components makes them approachable. Here's what to keep in mind:

• Look for perfect powers—like \( a^6 \) or \( b^6 \) in radicals—which can be simplified into base terms like \( a \) or \( b \).
• Separate terms under the radical into more straightforward pieces, making simplification a step-by-step process.

By recognizing patterns and identifying what can be factored or reduced, radical expressions become less daunting, transforming into an algebraic puzzle that you're fully capable of solving.

Constant Simplification

Dealing with constants in radical expressions is usually one of the first steps in simplification. Recognizing and managing constants reduces the complexity of the expression. Here's how to do it:

• Identify constants that are perfect powers matched with the radical degree, like 64, which matches the sixth root since \( 2^6 = 64 \).
• Extract the base of these constants from the radical, simplifying the expression to its greatest extent.

This process lays the groundwork for dealing with variables, helping you streamline and condense radical expressions right from the start. These steps form the backbone for tackling any similar problem effectively.