Problem 50 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \((4 x)^{3}\) when \(x=1\)

Step-by-Step Solution

Answer

The expression \((4x)^3\) when \(x=1\) evaluates to \(64\).

Step 1: Substitute Variable

Replace \(x\) in the expression \((4x)^3\) with the given value, which is \(1\). This operation leads to \((4*1)^3\).

Step 2: Apply BIDMAS rule

The BIDMAS rule refers to the order of operations in solving math problems. It states that: Brackets first, then Orders (ie Powers and Square Roots, etc.)), Division and Multiplication (left-to-right), Addition and Subtraction (left-to-right). Based on this rule, we first multiply \(4\) and \(1\) (as found inside the brackets). This yields \((4)^3\).

Step 3: Evaluate the Expression

Next, we calculate \(4^3\) (four to the power of three) which equals to \(64\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, knowing the order of operations is crucial. This set of rules dictates the sequence in which you should tackle different parts of a mathematical expression to ensure you get the correct result. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction, is often used to remember the hierarchy.

The critical point to remember is that multiplication and division, as well as addition and subtraction, are performed from left to right as they appear in the expression. This step is sometimes overlooked which could lead to incorrect answers. To prevent this, always start with the operations inside parentheses, followed by evaluating exponents. Once these are taken care of, move on to multiplication or division, whichever comes first as you read the expression from left to right, and finally, take on addition or subtraction.

Let's consider the given textbook exercise: \<4 x\>^3 when \(x=1\). Following PEMDAS, multiplication inside the parentheses should happen before addressing the exponent. Hence, the proper application of the order of operations is a vital skill for accurately solving algebraic expressions.

Substituting Values

To evaluate an algebraic expression for a specific value, we substitute the variables with the given numbers. This step involves replacing the variable with its given value and is essential for simplifying expressions and finding solutions.

In our exercise, the variable \(x\) is given the value \(1\). Substituting this value into our expression, \((4x)^3\), we replace \(x\) with \(1\), which simplifies the expression to \((4*1)^3\). With this action, we've taken the crucial step of turning an algebraic expression into a numerical one that we can then compute. It is important to substitute accurately, as errors in this step can lead to incorrect results.

After substitution, the simplified numerical expression can now be solved using the appropriate order of operations, which typically brings us to our next concept - exponents.

Exponents

Exponents represent how many times a number, known as the base, is multiplied by itself. They are an integral part of mathematical operations and appear frequently in algebra.

In notation, the exponent is the small superscript number next to the base. For example, in \(4^3\), \(4\) is the base, and \(3\) is the exponent, meaning that you multiply \(4\) by itself three times: \(4 \times 4 \times 4\). Therefore, \(4^3 = 64\).

Understanding the power of exponents allows you to simplify expressions swiftly and is particularly important for higher-level math where expressions can become quite complex. In the exercise provided, once we have substituted the variable and handled the operations within the parentheses, we then evaluate the exponent to finalize the solution.