Problem 21 - Algebra 1

Question

Use the power of a product property to simplify the expression. $$ \left(-2 x^{3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression of \((-2x^{3})^{3}\) is \(-8x^{9}\)

Step 1: Identify the base and the exponent

In the expression \((-2x^{3})^{3}\), the base inside the parentheses is -2 and x raised to power 3. Also, the whole expression inside the parentheses is raised to the power 3. Hence, the exponent is 3.

Step 2: Apply the power of a product property

Apply the power of a product property, which is \((ab)^m = a^m b^m\). Here, a is -2, b is \(x^{3}\), and m is 3. So, we get \((-2)^{3} (x^{3})^{3}\) as the simplified expression.

Step 3: Simplify the powers

Simplify \((-2)^{3}\) and \((x^{3})^{3}\). Since any number raised to the power of 3 means that number times itself twice, we get \(-8\) and \(x^{9}\).

Key Concepts

Exponentiation

Exponentiation is a powerful mathematical operation that involves raising a number, known as the base, to a particular power, called the exponent. This operation is expressed in the form \(a^n\), where \(a\) is the base and \(n\) is the exponent.

• The base is the number that is being multiplied.
• The exponent tells us how many times the base is used as a factor.

For example, in the expression \((-2x^3)^3\), \(-2x^3\) is the base and \(3\) is the exponent.Exponentiation is not limited to just numbers. It can be applied to algebraic expressions as well, where the base can be a simple variable like \(x\) or a more complex expression like \(-2x^3\). In such cases, understanding how to manipulate the powers of variables is crucial to simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and the operations of addition, subtraction, multiplication, and division. They are fundamental components of algebra and are used to model and solve real-world problems.

• Variables represent unknown values and are usually denoted by letters such as \(x\), \(y\), and \(z\).
• Constants are fixed numbers.
• Coefficients are numbers placed in front of the variables, indicating multiplication.

In the expression \((-2x^3)^3\):

• \(-2\) is a coefficient.
• \(x^3\) is a term comprising a variable \(x\) raised to the power 3.

Algebraic expressions often involve applying properties such as the power of a product to manipulate and simplify them, making it easier to work with or compare different expressions.

Simplifying Expressions

Simplifying expressions is a process used in algebra to transform complex expressions into simpler or more manageable forms without changing their values. This often involves using algebraic properties and rules, such as the power of a product property, which states that \((ab)^m = a^m b^m\).Here are the steps to simplify the expression \((-2x^3)^3\):

• Identify the base and the exponent: The base is \(-2x^3\), and the exponent is \(3\).
• Apply the power of a product property: This separates the base into its components, which are each then raised to the power of the exponent. This gives us \((-2)^3 (x^3)^3\).
• Simplify each component: Calculate \((-2)^3\) to get \(-8\), and \((x^3)^3\) to get \(x^9\).

Thus, \((-2x^3)^3\) simplifies to \(-8x^9\). By systematically applying rules and properties, algebraic expressions become less cumbersome, making it easier to understand and solve mathematical problems.