Problem 39 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(2 a^{5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(8a^{15}\).

Step 1: Apply the Power Rule

To simplify the expression \((2a^5)^3\), we use the power rule which states \((x^m)^n = x^{m \cdot n}\). We apply this to both the coefficient \(2\) and the variable \(a^5\).

Step 2: Simplify Coefficient

For the coefficient \(2\), we apply the power \(3\) resulting in \(2^3 = 8\).

Step 3: Simplify the Variable

For the variable part \((a^5)^3\), using the power rule, it becomes \(a^{5 \cdot 3} = a^{15}\).

Step 4: Combine the Results

Combine the simplified parts: \(8\) and \(a^{15}\). The expression simplifies to \(8a^{15}\).

Key Concepts

Understanding the Power of a Product

The power of a product rule is a handy tool when you need to simplify expressions that involve multiplication inside parentheses and an exponent outside. When you see an expression like \((ab)^n\), the power of the product rule tells us that each element inside the parentheses is raised to the power \(n\). In other words, \((ab)^n = a^n \cdot b^n\). This rule allows you to distribute the outer exponent to each factor inside.
For our original exercise, \((2a^5)^3\), we can apply this rule directly. Here, we think of it as \(2 \cdot a^5\) raised to the third power. By distributing the power of \(3\), we apply it to both the \(2\) and the \(a^5\), resulting in \(2^3 \cdot (a^5)^3\). This simplification process helps in managing complex expressions by breaking them down into smaller, manageable parts.

Mastering the Quotient Rule

While our original exercise does not directly involve a quotient, understanding the quotient rule is pivotal for mathematical expressions that involve division. The quotient rule is similar to the power of a product rule but applies to fractions instead. If you encounter an expression like \(\left( \frac{a}{b} \right)^n\), the quotient rule states that you apply the power \(n\) to both the numerator and the denominator separately. Therefore, \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\).
This rule essentially allows you to handle the power on each part of a fraction individually, simplifying complex rational expressions effectively. It's essential to correctly apply this rule when both a numerator and denominator are raised to a power since errors here can lead to incorrect simplifications.

• Ensure each part of the fraction receives the exponent.
• Simplify each part before attempting to combine for clarity.

Techniques for Simplifying Expressions

Simplifying expressions is all about making them easier to work with by reducing complexity as much as possible. This involves applying various algebraic rules, like the power rule, to make expressions easier to interpret and use.
In our exercise, we simplified \((2a^5)^3\) using the power rule. We first simplified each component: \(2^3\) leads to \(8\) and \((a^5)^3\) gives us \(a^{15}\). Then, we combined them to form \(8a^{15}\). This process makes it easier to evaluate or use the expression in equations.
Here are some quick tips for simplifying expressions effectively:

• Identify and apply rules of exponents, like the power rule, accurately.
• Ensure all like terms are combined to avoid redundant calculations.
• Always double-check your simplification by going over each step to confirm correctness.

By doing this, the expression is not only simpler but also clearer and less prone to errors in further mathematical operations.