Problem 14 - 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: Apply the Power of a Power Property

The expression is given as \( \left(-3x^{3}\right)^{2} \). According to the power of a power property, \((a^m)^n = a^{m\cdot n}\). Thus, \((x^3)^2 = x^{3 \cdot 2} = x^6\).

Step 2: Distribute the Exponent

Now apply the exponent to each factor inside the parentheses: \((-3)^2 \cdot (x^3)^2\). We have already simplified \((x^3)^2\) to \(x^6\).

Step 3: Calculate the Power of Negative Three

Calculate \((-3)^2\). This simply means multiplying \(-3\) by itself: \((-3) \times (-3) = 9\).

Step 4: Combine the Results

Now combine the results from Step 1 and Step 3: \(9 \cdot x^6\). Thus, the simplified expression is \(9x^6\).

Key Concepts

Power of a Power Property

Understanding the power of a power property is essential when working with exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. The general formula is written as

• \((a^m)^n = a^{m \cdot n}\),

where \(a\) is the base and \(m\) and \(n\) are exponents.
This rule simplifies expressions and makes calculations quicker by transforming multiple layers of exponents into a single one. For example, in the expression \((x^3)^2\), you apply the power of a power property by multiplying the exponents: \(3 \times 2 = 6\). Thus, \((x^3)^2\) simplifies to \(x^6\).
This concept is particularly useful in algebra and calculus, as it helps break down complex expressions into manageable parts.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form without changing their value. It's like cleaning up your work to make it look neat and clear.
When you simplify, you're making an expression as compact and straightforward as possible. For example, in the expression \(\left(-3x^3\right)^2\), you need to manage each piece of the expression separately, using exponent rules and basic arithmetic operations.

• First, apply the exponent to the inner expression using the power of a power rule: \((x^3)^2 = x^6\).
• Next, deal with the numerical coefficient: Calculate \((-3)^2\) which simplifies to \(9\).
• Finally, combine these bits to get the simplified expression: \(9x^6\).

By simplifying expressions, you make them easier to work with, especially in more complex algebraic operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operators (like addition or multiplication). These expressions are the building blocks of algebra and can represent any number of values.
In our example, \((-3x^3)^2\) is an algebraic expression that includes a coefficient (-3) and a variable term \(x^3\).

• The coefficient is the numerical part that multiplies the variable.
• The variable is \(x\), raised to the power of 3 in this instance.

Algebraic expressions can be manipulated in many ways, such as adding, subtracting, multiplying, dividing, or applying exponent rules. These manipulations make solving and simplifying mathematical problems feasible. Understanding how to handle algebraic expressions enables you to solve equations and model real-world situations using math.