Problem 93 - Elementary Algebra

Question

Problem: Simplify: \(\left(6 n^{3}\right)^{2}\) $$ \text { Incorrect Answer: } \begin{aligned} &\left(6 n^{3}\right)^{2} \\ &=6^{2}\left(n^{3}\right)^{2} \\ &=36 n^{5} \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \(36 n^6 \).

Step 1: Understand the Problem

The goal is to simplify the expression \(\left(6 n^{3}\right)^{2}\). Be aware that the incorrect answer is \(36 n^{5}\).

Step 2: Use the Power of a Product Rule

The power of a product rule states \( (ab)^m = a^m b^m \). Apply this rule to the expression \(\left(6 n^{3}\right)^{2}\).

Step 3: Separate and Simplify

Using the power of a product rule: \(\left(6 n^{3}\right)^{2} = 6^2 \left(n^{3}\right)^{2}\).

Step 4: Simplify Each Component

Calculate each part of the expression separately: \( 6^2 = 36\) and \(\left(n^{3}\right)^{2} = n^{3 \cdot 2} = n^{6}\).

Step 5: Combine the Simplified Parts

Combine the results from Step 4: \( 6^2 \left(n^{3}\right)^{2} = 36 n^{6} \).

Step 6: Final Answer

The simplified form of the expression is \(36 n^6 \).

Key Concepts

Power of a Product Rule

When simplifying expressions that involve both multiplication and exponentiation, the power of a product rule is very handy. This rule states that for any numbers or variables, if we have a product raised to a power, like \((ab)^m\), the expression can be separated as \((a^m b^m)\). Here’s a step-by-step breakdown:

• First, identify the product inside the parentheses. In our exercise, it’s \(6n^3\).
• Next, apply the power to each part of the product separately. For \((6n^3)^2\), this means applying the power of 2 to both 6 and \(n^3\).

This rule simplifies complex expressions by breaking them down. We see that \( (6n^3)^2 = 6^2 (n^3)^2 \). The power rule allows us to manage exponents easily, making the expression simpler to work with.

Exponentiation

Exponentiation refers to the operation of raising a base to a power. For example, \(a^m\) means a multiplied by itself m times. It is a way to denote repeated multiplication. Here are key points:

• The base is the number being multiplied. In the exercise, 6 and \(n^3\) are bases.
• The exponent is the number that indicates how many times the base is used in the multiplication. For instance, in \((6n^3)^2\), the exponent 2 is applied to both 6 and \(n^3\).

Apply exponentiation to each part:

• For 6 raised to the power of 2, calculate \(6^2 = 6 \times 6 = 36\).
• For \(n^3\) raised to the power of 2, use the rule \((n^b)^m = n^{b \times m}\). Calculate \((n^3)^2 = n^{3 \times 2} = n^6\).

This makes exponentiation crucial in simplifying expressions, transforming a complex problem into easier calculations.

Simplifying Expressions

Simplifying expressions involves reducing them to a simpler form without changing their value. The goal is to make them easier to understand and work with. For our problem, we follow these steps:

• First, use the power of a product rule to distribute the exponent. From \((6n^3)^2\), we get \(6^2 (n^3)^2\).
• Next, simplify each part individually. Calculate the powers: \( 6^2 = 36 \) and \((n^3)^2 = n^6 \).
• Finally, combine the simplified parts to get the final answer. Therefore, \( (6^2)(n^6) = 36n^6 \).

By following these steps systematically, we transform a complex algebraic expression into its simplest form. Simplifying correctly ensures accurate results and helps in solving equations more efficiently. Always verify every step to avoid common mistakes.