Problem 40 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$-\sqrt[6]{64 a^{12} b^{8}}$$

Step-by-Step Solution

Answer

Simplified expression: \(-2a^2 b^{4/3}.\)

Step 1: Simplify the Numerical Part

Start by simplifying the numerical part of the radical expression, \(-\sqrt[6]{64}\). Since \(2^6 = 64\), we have \(-\sqrt[6]{64} = -2.\)

Step 2: Apply the Radical to the Power of a

For the variable \(a^{12}\), use the property of radicals that states \(\sqrt[n]{a^m} = a^{m/n}\). The expression becomes \(\sqrt[6]{a^{12}} = a^{12/6} = a^2.\)

Step 3: Apply the Radical to the Power of b

Simplify \(\sqrt[6]{b^8}\) using the same property: \(\sqrt[6]{b^8} = b^{8/6} = b^{4/3}.\) This can also be written as \(b^{1 + 1/3}.\)

Step 4: Combine the Results

Multiply the simplified parts together, combining steps where possible. Thus, the expression simplifies to: \( -2 a^2 b^{4/3}.\)

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. A radical expression can contain numbers or variables under these roots. To write a radical, we use the radical sign \( \sqrt{} \) for a square root or \( \sqrt[n]{} \) for an nth root. In this case, the given expression is \(-\sqrt[6]{64 a^{12} b^{8}}\), which is a sixth root radical. It's like asking what value, when raised to the sixth power, will give the expression inside the root.
Radical expressions are a foundational part of algebra as they often come up in equations that involve powers and roots. Simplifying these expressions makes them easier to work with and forms the basis for further algebraic manipulations.

Properties of Radicals

Understanding the properties of radicals helps in simplifying complex radical expressions. These properties are rules that apply to roots and can be invaluable for reducing the form of an expression under a radical sign. Here are a few crucial properties:

• If \( a^n = b \), then \( \sqrt[n]{b} = a \).
• \( \sqrt[n]{a^m} = a^{m/n} \): This property allows us to convert a radical expression into an expression with fractions as exponents. It's particularly useful when dealing with variables.
• Radicals can be multiplied: \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} \).
• Simplification aims to make the number inside the radical as small as possible. This sometimes involves factoring the number or result into a simpler square or higher power.

By carefully applying these properties, we simplify radical expressions step-by-step, reducing their complexity while maintaining their equivalence.

Simplifying Expressions

Simplifying radical expressions involves a step-by-step approach to break down the expression into more manageable parts. The objective is to reduce the radical to its simplest form by identifying and removing perfect powers.
Let's look at our problem: start with the numerical part. We noticed that 64 is a perfect sixth power, since \( 2^6 = 64 \), thus \( \sqrt[6]{64} = 2 \). The next task is simplifying variables under the radical.
Follow these easy steps to simplify the variables:

• For \( a^{12} \), apply \( \sqrt[n]{a^m} = a^{m/n} \): \( \sqrt[6]{a^{12}} = a^{12/6} = a^2 \).
• Understand the expression \( \sqrt[6]{b^8} = b^{8/6} = b^{4/3} \). Here, the exponent is a fraction, meaning \( b^{4/3} \) stands for \( b \) to the power of 1 plus an additional third.

Finally, gather all simplified parts. Combine them, remembering any negative or multipliers initially found. Thus, the expression \( -\sqrt[6]{64 a^{12} b^{8}} \) neatly simplifies to \( -2 a^2 b^{4/3} \). This effectively reduces the complexity of working with the radical in equations or further study.