Problem 24 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(64 a^{12}\right)^{5 / 6}$$

Step-by-Step Solution

Answer

The simplified expression is \(32a^{10}\).

Step 1: Apply the Power of a Power Property

The expression \((64a^{12})^{\frac{5}{6}}\) uses the property that \( (x^m)^n = x^{m \cdot n} \). Apply this to both the constant and variable parts separately.

Step 2: Simplify the Exponent of the Constant

Rewrite \(64^{\frac{5}{6}}\) by first expressing 64 as a power of a smaller base. Since \(64 = 2^6\), the expression becomes \( (2^6)^{\frac{5}{6}} = 2^{6 \cdot \frac{5}{6}} = 2^5 \).

Step 3: Simplify the Exponent of the Variable

Now, apply the same property to the variable part: \(a^{12}\cdot \frac{5}{6} = a^{\frac{12 \cdot 5}{6}} = a^{10}\).

Step 4: Combine the Simplified Parts

Combine the simplified constant and variable expressions: \(2^5 a^{10}\). Calculate \(2^5 = 32\), giving the final simplified expression as \(32a^{10}\).

Key Concepts

Power of a Power Property

When working with exponents, one of the key properties you'll encounter is the "Power of a Power Property." This property is useful for simplifying expressions where an exponent is raised to another power. The formula for this property is

• \( (x^m)^n = x^{m \cdot n} \)

This formula tells us that when you have a power raised to another power, you can simplify by multiplying the exponents together.
In practice, this means if you see an expression like \((64a^{12})^{\frac{5}{6}}\), you will apply the rule to both the base of the number and the variable separately. The key is to multiply their exponents, which allows us to simplify the expression effectively.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. Simply put, it's like multiplying a number by itself a certain number of times.

• For example, \(2^3\) means \(2 \times 2 \times 2 = 8\).

When dealing with more complex bases such as \(64\), which can be expressed as a power of a smaller base like \(2^6\), exponentiation helps in further simplifying expressions. In our exercise, 64 is rewritten as \((2^6)^{\frac{5}{6}}\), allowing us to apply the power of a power property by calculating \( 2^{6 \cdot \frac{5}{6}} = 2^5 \).
This simplification step is crucial because it breaks down what seems complicated into something manageable.

Simplifying Expressions

The process of simplifying expressions is about reducing them to their simplest form, making them easier to understand or solve. It combines various algebraic rules and properties, such as the distributive property, the power of a power property, and basic arithmetic simplifications.
In the exercise \((64a^{12})^{\frac{5}{6}}\), once exponentiation has been simplified using the power of a power property and rewritten bases, we explore how to simplify both parts of the expression:

• For the constant: \(64\) simplifies to \(2^5 = 32\).
• For the variable: \(a^{12 \cdot \frac{5}{6}} = a^{10}\).

Finally, combining these gives us \(32a^{10}\), the simplest form of the original expression. This process utilizes the rules of algebra to peel back layers of complexity step by step.