Problem 44 - Elementary Algebra

Question

For the following problems, select a number (or numbers) to show that \((5 x)^{2}\) is not generally equal to \(5 x^{2}\).

Step-by-Step Solution

Answer

Answer: No, they are not generally equal.

Step 1: Select a number for x

Choose a number for x, in this case, let's pick x = 2.

Step 2: Calculate \((5x)^{2}\)

Plug the value of x = 2 into the first expression and calculate: \((5(2))^{2} = (10)^{2} = 100\).

Step 3: Calculate \(5x^2\)

Now, plug the value of x = 2 into the second expression and calculate: \(5(2)^{2} = 5(4) = 20\).

Step 4: Compare the results

We found that \((5x)^{2} = 100\) and \(5x^2 = 20\) when x = 2. Since \(100 \neq 20\), it clearly shows that the two expressions are not generally equal.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. When you see a number or variable raised to an exponent, like in \(x^2\), it means that number or variable is multiplied by itself. For instance, \(x^2 = x \times x\). Working with exponents is a fundamental aspect of algebra, allowing us to handle large numbers or simplify expressions.

It's important to differentiate between applying exponents to a single term and to an entire expression. In the exercise problem, we explored this difference: \((5x)^2\) versus \(5x^2\).

• For \((5x)^2\), the whole expression \(5x\) is squared, yielding \((5x) \times (5x) = 25x^2\).
• In the case of \(5x^2\), only \(x\) is squared, which means we simply have \(5 \times (x \times x) = 5x^2\).

Understanding this distinction is crucial for correctly evaluating and comparing expressions that involve exponents. Using numerical values, like we did with \(x = 2\), helps to visualize the differences.

Distributive Property

The distributive property is a key algebraic rule that allows you to multiply a sum by distributing the multiplication over each addend within the parentheses.

For any numbers \(a\), \(b\), and \(c\), it is expressed as: \[a(b + c) = ab + ac\] This property can also extend to expressions that contain variables and exponents. While the distributive property doesn't directly apply to the exercise at hand, understanding how multiplication interacts with grouped terms is essential.

When comparing \((5x)^2\) and \(5x^2\), one might be tempted to apply the distributive property incorrectly. It’s necessary to grasp that you can’t distribute exponents over multiplication in the way you do with the distributive property. Instead, you need to evaluate each component of the expression.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division are building blocks in algebra. These operations allow us to manipulate expressions to simplify, expand, or solve them.

In this exercise, multiplication plays the primary role, especially while differentiating between \((5x)^2\) and \(5x^2\).

• In \((5x)^2\), you multiply the whole term \(5x\) by itself, which requires expanding via multiplication: \((5x) \times (5x)\).
• Conversely, in \(5x^2\), only the \(x^2\) indicates that \(x\) is squared before multiplying by 5.

This exercise emphasizes the importance of performing operations in the correct order and correctly interpreting notational cues in expressions. Adhering to the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) ensures precise and accurate solutions. It also highlights the nuances of how terms can be grouped and treated during calculations.