Problem 93 - Elementary Algebra

Question

Choose value for \(x\) to show that a. \((4 x)^{2}\) is not always equal to \(4 x^{2}\). b. \((4 x)^{2}\) may be equal to \(4 x^{2}\).

Step-by-Step Solution

Answer

Question: Provide an example demonstrating that (4x)² is not always equal to 4x², and another example showing when they may be equal. Answer: When x = 1/2, (4x)² = 4 and 4x² = 1, which shows that these expressions are not always equal. However, when x = 0, both (4x)² and 4x² equal 0, demonstrating that they may be equal in some cases.

Step 1: Choose a value for \(x\) that makes the expressions different

A general suggestion for this type of problem is to choose a number that introduces a clear difference between the two expressions. A good choice would be \(x = \frac{1}{2}\).

Step 2: Evaluate both expressions using the chosen value

Evaluate the two expressions with the chosen value of \(x\): 1. \((4x)^2 = \left(4\left(\frac{1}{2}\right)\right)^2 = (2)^2 = 4\) 2. \(4x^2 = 4\left(\frac{1}{2}\right)^2 = 4\left(\frac{1}{4}\right) = 1\)

Step 3: Compare the results

By comparing the results of both expressions, we see that they are different. \((4x)^2 = 4 \neq 1 = 4x^2\) b. Proving \((4x)^2\) may be equal to \(4x^2\)

Step 1: Choose a value for \(x\) that makes the expressions equal

The simplest choice for this step would be \(x = 0\) since any number times zero is still zero.

Step 2: Evaluate both expressions using the chosen value

Evaluate the two expressions with the chosen value of \(x\): 1. \((4x)^2 = (4(0))^2 = (0)^2 = 0\) 2. \(4x^2 = 4(0)^2 = 4(0) = 0\)

Step 3: Compare the results

By comparing the results of both expressions, we see that they are equal. \((4x)^2 = 0 = 0 = 4x^2\) In conclusion, we have shown that there exists a value for \(x\) that makes \((4x)^2\) not equal to \(4x^2\) and another value that makes them equal.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They can represent a wide range of situations in mathematics and are a fundamental part of algebra. For instance:

• The expression \(4x\) represents 4 times a variable \(x\).
• Adding an exponent, like in \((4x)^2\), changes how we manipulate the expression.

Variables in expressions allow us to evaluate outcomes by substituting different numerical values. This makes algebraic expressions versatile in solving mathematical problems. When we change a variable, it can significantly affect the result of the expression, which is precisely what we see in the given exercise.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. In algebraic terms, exponentiation is used to raise a number or expression to a particular power. Here's how it works:

• The base is the number or expression being multiplied by itself.
• The exponent indicates how many times the base is multiplied.

For example, in the expression \[(4x)^2 = (4x) imes (4x)\]Each part of the expression is multiplied twice. This distinction becomes critical when dealing with expressions like \((4x)^2\) versus \(4x^2\). It's crucial to recognize that \[4x^2 = 4 imes (x imes x)\]which is not the same as multiplying the whole expression \(4x\) by itself. Understanding these differences helps clarify why certain values of \(x\) result in different outcomes for these expressions.

Equivalent Expressions

Equivalent expressions are different expressions that have the same value for all values of variables involved. To determine equivalency, substitute values and evaluate to see if the expressions yield the same result. A critical point in the exercise is demonstrating when two expressions are equivalent or not.
For example:

• Choosing \(x = 0\), both expressions \((4x)^2\) and \(4x^2\) resolve to \(0\), making them equivalent.
• On the other hand, choosing \(x = \frac{1}{2}\) results in distinct values for the expressions, showing they are not equivalent at this point.

Equivalent expressions are fundamental in algebra as they allow flexibility in computation and simplification of expressions. They are particularly useful in solving equations, where different forms of an expression can lead to the same solution, enhancing understanding and application in mathematics.