Problem 23 - A Graphical Approach to College Algebra

Question

Simplify each expression. Assume that all variables represent positive real numbers. $$\left(27 x^{6}\right)^{2 / 3}$$

Step-by-Step Solution

Answer

The simplified expression is \(9x^4\).

Step 1: Apply the Power of a Power Rule

When a power is raised to another power, you can multiply the exponents. Here, the expression is \( (27 x^6)^{2/3} \). This can be expressed as \( 27^{2/3} \cdot x^{(6 \cdot 2/3)} \).

Step 2: Simplify the Exponents

First, calculate \(6 \cdot 2/3 = 4\). Therefore, \(x^{6 \cdot 2/3} = x^4\).

Step 3: Evaluate the Cube Root and Square

The expression \(27^{2/3}\) can be interpreted as \((27^{1/3})^2 \). The cube root of 27 is 3, so \(27^{1/3} = 3\). Therefore, \((27^{1/3})^2 = 3^2 = 9\).

Step 4: Combine the Results

Now that we have simplified \(27^{2/3} = 9\) and \(x^{6 \cdot 2/3} = x^4\), we combine them to get the final simplified expression: \(9x^4\).

Key Concepts

Understanding the Power of a Power Rule

The power of a power rule is a valuable tool in algebra. It helps simplify expressions with exponents efficiently. When you raise an exponent to another exponent, you multiply the exponents together. This means if you have an expression like

• \((a^m)^n\),

it becomes

• \(a^{m \cdot n}\).

Let's take our example,

• \((27x^6)^{2/3}\).

We can distribute the exponent to each part.
For the variable part, it becomes \(x^{6 \cdot 2/3}=x^4\). This rule makes it easier to break down complex expressions into manageable parts.

Exploring Exponentiation

Exponentiation is the process of raising a number to a power. It’s a foundational concept in algebra. When you see something like

• \(a^b\),

it means multiplying \(a\) by itself \(b\) times.
For example, \(3^2 = 3 \times 3 = 9\).

In the exercise, we had \(27^{2/3}\). It’s useful to think of it in steps:

• First, find the cube root: \(27^{1/3}\).
• The cube root of 27 is 3, so \(27^{1/3} = 3\).
• Next, square the result: \(3^2 = 9\).

Using these small steps helps simplify even those tricky fractional exponents.

Simplifying Expressions Effectively

Simplifying expressions is all about making mathematics easier to handle. The goal is to reduce an expression to its simplest form. Let's see how we did it with

• \((27x^6)^{2/3}\).

First, apply the power of a power rule to simplify the variable part:

• \(x^{6 \cdot 2/3} = x^4\).

Next, tackle the number part:

• \(27^{2/3} = 9\) (as we found through cube rooting and then squaring).

Finally, combine these results to get the simple expression—all wrapped up as \(9x^4\). Simplifying helps in solving equations and understanding their underlying structure more clearly.