Problem 7 - Elementary Algebra

Question

Select a number to show that \((5 x)^{2}\) is not always equal to \(5 x^{2}\).

Step-by-Step Solution

Answer

Explain using an example. Answer: No, \((5x)^2\) and \(5x^2\) are not always equal. For example, when x = 2, \((5x)^2 = (10)^2 = 100\), whereas \(5x^2 = 5 \cdot (2)^2 = 20\). Since the two values are not equal, this demonstrates that \((5x)^2\) is not always equal to \(5x^2\).

Step 1: Select a number for x

Let's choose a number for x to use in our expressions. For this example, let's use x = 2.

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

Now, we will substitute x with our chosen value in the expression \((5x)^2\). \((5 \cdot 2)^2 = (10)^2 = 100\) So, \((5x)^2\) is equal to 100 when x = 2.

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

Next, we will substitute x with our chosen value in the expression \(5x^2\). \(5 \cdot (2)^2 = 5 \cdot 4 = 20\) So, \(5x^2\) is equal to 20 when x = 2.

Step 4: Compare the two results

Let's compare the results of \((5x)^2\) and \(5x^2\) when x = 2. \((5x)^2 = 100\) and \(5x^2 = 20\) Since the two values are not equal, we have demonstrated that \((5x)^2\) is not always equal to \(5x^2\).

Key Concepts

Exponents

Exponents are a shorthand way of indicating that a number, known as the base, is multiplied by itself a certain number of times. In the expression \((5x)^2\), the exponent \(2\) tells us to multiply \(5x\) by itself, which means \((5x) imes (5x) = 25x^2\). Similarly, in the expression \(5x^2\), the exponent \(2\) affects only the \(x\), indicating that \(x\) is multiplied by itself: \(x imes x = x^2\). Here, the base \(5\) only multiplies the result of \(x^2\), resulting in \(5 imes x^2 = 5x^2\). This key difference in how the exponent is applied to the base entails vastly different outcomes, as we've seen with our example where \(x = 2\): \((5 imes 2)^2 = 100\) versus \(5 imes 2^2 = 20\). Exponents play a crucial role in determining how expressions are evaluated.

Order of Operations

The order of operations is a standardized procedure in mathematics used to ensure expressions are interpreted and solved consistently. When faced with expressions involving multiple operations—addition, subtraction, multiplication, division, and exponentiation—the order of operations must be applied. The widely remembered acronym PEMDAS represents:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, the distinction between \((5x)^2\) and \(5x^2\) relies on correctly applying this order. In \((5x)^2\), everything within the parentheses is handled first, followed by raising the result to the power of \(2\). Alternatively, \(5x^2\) instructs us to first compute \(x^2\), then multiply the outcome by \(5\). Ignoring the order of operations could lead to entirely different and often incorrect results.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operations, and sometimes grouping symbols like parentheses. They are constructed by following the rules of arithmetic to represent numbers or relationships between numbers. In the exercise above, both \((5x)^2\) and \(5x^2\) are examples of algebraic expressions. Though they look similar, the presence of parentheses in \((5x)^2\) changes the way the expression is calculated.
Expressions often form the foundation of equations and functions, which are used to model and solve real-world problems. Understanding the building blocks of mathematical expressions, like learning how to manipulate variables and apply operations correctly, is vital for success in algebra and all future math courses. By correctly evaluating an expression step-by-step and applying rules such as the order of operations, students can develop proficiency and confidence in handling complex mathematical tasks.