Problem 57 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(2 x^{3} \cdot(-3 x)^{2}\)

Step-by-Step Solution

Answer

The simplified expression of the given \(2 x^{3} \cdot(-3 x)^{2}\) is \(18x^{5}\).

Step 1: Apply Exponentiation Rule to (-3x)^2

The exponentiation rule \((a^{m})^n = a^{m\cdot n}\) is applied to \((-3x)^2\). It results in \((-3)^2 \cdot x^{2\cdot1}\). Remember that both -3 and \(x^{1}\) need to be squared.

Step 2: Calculate the Squares

Calculate the squares of -3 and \(x^{1}\). That is, \((-3)^2 = 9\) and \(x^{2\cdot1} = x^{2}\). This simplifies the expression to \(2x^{3}\cdot9x^{2}\).

Step 3: Simplify the Mathematical Expression

Use the exponentiation rule to simplify the expression \(2x^{3}\cdot9x^{2}\). According to the exponentiation rule \(a^{m} \cdot a^{n} = a^{m+n}\), the exercise results in \(2\cdot9 \cdot x^{3+2}\).

Step 4: Calculate the Resulting Expression

Finally, calculate \( 2\cdot9 \cdot x^{3+2}\) which equals to \(18x^{5}\).

Key Concepts

Exponentiation Rule

Exponentiation is a mathematical operation, involving numbers raised to a power. It's fundamental when dealing with expressions containing exponents. In this exercise, first we look at how to apply the exponentiation rule effectively.

Consider the expression \((-3x)^2\). To exponentiate, you need to apply the power to both components inside the parentheses. This expression can be expanded using the rule \((a^m)^n = a^{m \cdot n}\). This means both the components, \(-3\) and \(x\), are raised to the second power individually:

• \((-3)^2\)
• \(x^{2 \cdot 1}\)

Thus, \((-3x)^2\) becomes \(9x^2\), because \((-3)^2 = 9\) and \(x^{2 \cdot 1} = x^2\). This clarity in separating and squaring components is crucial for accurate simplification.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical operations to their simplest form. It's a useful skill that helps in solving equations more easily. In the given exercise, once we apply the exponentiation rule and find that \((-3x)^2\) equates to \(9x^2\), we can then multiply it with the term \(2x^3\).

Think of terms with variables as items you can combine. Here, you combine the coefficients (numbers) separately from the variables:

• The coefficients: \(2 \cdot 9 = 18\)

Next, tackle the variables using exponentiation:

• \(x^3 \cdot x^2 = x^{3+2} = x^5\)

The concept hinges on accurately combining like terms, ensuring each part of the expression is simplified before combining. This process unveils the simplified expression \(18x^5\).

Polynomial Multiplication

Polynomial multiplication is the process of multiplying polynomials together, such as monomials in our case. Mastery of polynomial multiplication ensures you can handle any variable raised to a power, combined with different coefficients.

In this exercise, after simplifying the expression \((2x^3)\cdot(9x^2)\) and applying the exponentiation rule, we're left with pure polynomial multiplication.

First, focus on multiplying the coefficients \(2\) and \(9\) which gives \(18\). Then carefully handle the \(x\) terms. The multiplication of like bases \(x^3\) and \(x^2\) requires adding their exponents, thus becoming \(x^{3+2}\).

By following these simple steps, the polynomial multiplication ends with the expression \(18x^5\). Breaking down each component separately before multiplying ensures clarity and a correct final solution.