Problem 13 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(-\left(3 x^{3}\right)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(-9x^6\).

Step 1: Understand the Expression

The given expression is \(-\left(3 x^{3}\right)^{2}\). This involves both a negative sign and an exponentiation operation. Our goal is to simplify it.

Step 2: Apply the Power Rule

Use the power rule: \((a^m)^n = a^{m \cdot n}\). Apply this to the expression to multiply the exponents, resulting in \(-\left(3^2 \cdot (x^3)^2\right)\).

Step 3: Calculate the Powers

Calculate each power separately:\(3^2 = 9\) and \((x^3)^2 = x^{3 \cdot 2} = x^6\). Thus, the expression becomes \(-9x^6\).

Step 4: Simplify Result

The expression \(-9x^6\) is already in its simplest form since there are no further operations to perform.

Key Concepts

Simplifying Expressions

When working with algebraic expressions, especially those involving exponents, simplifying them means reducing the expression to the most basic form without changing its value.
To simplify expressions like \(-\left(3 x^{3}\right)^{2}\), follow these steps:

• Identify each component of the expression—this includes coefficients (numbers in front), variables, and exponents.
• Apply relevant rules and operations, such as distributing exponents over products or combining like terms when necessary.
• Use arithmetic operations to simplify numerical parts.

Simplified expressions are easier to work with, as they require less computation in further operations or when substituting numerical values.

Power Rule

In algebra, the power rule is a key tool for simplifying expressions that involve exponents. This rule states that when you raise a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
Here are the steps for applying the power rule:

• Identify the base and the exponents. In our example, \(3x^3\) is the base, and \(2\) is the exponent applied to this entire base.
• Apply the power rule to each part of the base separately. So, for \(3\), we calculate \(3^2\), and for \(x^3\), apply \((x^3)^2\) which becomes \(x^{3 \cdot 2}\).
• Combine the results of these operations, noting that each element within the parentheses is raised to the cumulative power derived from the powers applied.

With the power rule, you simplify expressions quickly by managing and reducing exponents effectively.

Negative Sign Handling

Dealing with negative signs in expressions is crucial for accurate simplification. In our expression, the negative sign in front \(-\left(3 x^{3}\right)^{2}\) initially might suggest an additional step, but it just affects the sign of the final product.
Tips for handling a negative sign:

• Keep the negative sign separate until the other operations are complete. In our case, compute \((3 x^3)^2\) first.
• Multiply the simplest result by \(-1\) afterwards. For instance, once you've calculated \((3x^3)^2\) as \(9x^6\), the negative sign makes it \(-9x^6\).
• Ensure not to distribute the negative sign inside parentheses when not required, as it can lead to errors.

This approach guarantees that your final expression retains the proper sign, crucial for arithmetic accuracy and application.