Problem 45 - Elementary Algebra

Question

For the following problems, select a number (or numbers) to show that \((7 x)^{2}\) is not generally equal to \(7 x^{2}\).

Step-by-Step Solution

Answer

Question: Show that \((7x)^2\) is not generally equal to \(7x^2\) using an example. Answer: When we choose x = 2, we find that \((7x)^2 = 196\) while \(7x^2 = 28\). Since these values are different, we can conclude that \((7x)^2\) is not generally equal to \(7x^2\).

Step 1: Select the value of x

We can choose any number for x, but to keep things simple, let's choose x = 2.

Step 2: Calculate \((7x)^2\) for x = 2

Now, let's calculate \((7x)^2\) for the chosen value of x (x = 2). Plug in the value of x into the formula: \((7 \times 2)^2\).

Step 3: Simplify the expression \((7x)^2\) for x = 2

To simplify \((7 \times 2)^2\), first multiply the numbers inside the parentheses: \((14)^2\). Now, square the number inside the parentheses: \(196\).

Step 4: Calculate \(7x^2\) for x = 2

Now, calculate \(7x^2\) for the chosen value of x (x = 2). Plug in the value of x into the formula: \(7 \times (2)^2\).

Step 5: Simplify the expression \(7x^2\) for x = 2

To simplify \(7 \times (2)^2\), first square the number inside the parentheses: \(7 \times 4\). Now, multiply the numbers outside the parentheses: \(28\).

Step 6: Compare the results of \((7x)^2\) and \(7x^2\) for x = 2

We found that \((7x)^2 = 196\) and \(7x^2 = 28\) when x = 2. Since the two expressions yield different results, we can conclude that \((7x)^2\) is not generally equal to \(7x^2\).

Key Concepts

Exponents

Exponents play a critical role in algebra and are often depicted as a small number written to the upper right of a base number. This exponent tells us how many times to multiply the base number by itself. For example, in the expression

• \((a^2)\), the base \(a\) is multiplied by itself once, resulting in \(a \times a = a^2\).
• When you see something like \((7x)^2\), this means that the entire product \(7x\) is raised to the power of 2. Thus, \((7x)^2 = (7x) \times (7x)\), which expands further to \(7 \times x \times 7 \times x\), ultimately simplifying to \(49x^2\).

Exponents clarify the magnitude of numbers by compressing repeated multiplication into a more manageable form. They are essential in simplifying expressions and solving equations. When dealing with exponents in expressions, especially like \((7x)^2\) and \(7x^2\), understanding their application becomes crucial in verifying if two expressions are equivalent.

Simplification

Simplification in algebra involves reducing expressions to their most basic form. This is accomplished by performing the necessary arithmetic operations and applying the order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right). Let's take a closer look at our example.For

• \((7x)^2\), a good simplification process starts with multiplying the terms inside the parentheses before applying the exponent:
• First, multiply \(7 \times 2\) to get \(14\).
• Next, square the result: \(14^2 = 196\).
• In contrast, simplifying \(7x^2\) begins with squaring \(x\):
• Square \(2\) to get \(4\).
• Then, multiply \(7\) by this result: \(7 \times 4 = 28\).

The distinct order in which operations are carried out affects the final result, emphasizing the need for careful simplification to avoid errors.

Expression equality

Expression equality is a key concept in algebra that involves determining whether two expressions have the same value for all values of the variable(s) involved. Identifying expression equality is fundamental when solving equations or verifying algebraic identities.In the original problem, you're tasked with showing that \((7x)^2\) is not generally equal to \(7x^2\). After performing computations for a specific example where \(x = 2\):

• We discovered that \((7x)^2\) gives us \(196\).
• Meanwhile, \(7x^2\) results in \(28\).

Since \(196 eq 28\), it highlights that simply having similar-looking terms does not imply equality without considering the order of operations. Therefore, the two expressions are not inherently equal, as they yield different results for the same value of \(x\). Understanding how expressions relate or differ from one another is essential for solving more complex algebraic problems.