Problem 70 - Algebra 2

Question

Evaluate each expression if \(a=-5, b=0.25, c=\frac{1}{2},\) and \(d=4 .(\text { lesson } 1-1)\) $$ b+3(a+d)^{3} $$

Step-by-Step Solution

Answer

The evaluated expression is -2.75.

Step 1: Substitute Variables

Given the expression \(b+3(a+d)^{3}\), substitute the specified values of \(a = -5\), \(b = 0.25\), \(c = \frac{1}{2}\), and \(d = 4\). The expression becomes: \(0.25 + 3(-5 + 4)^{3}\).

Step 2: Simplify Inside the Parentheses

Calculate \((-5 + 4)\), which simplifies to \(-1\). Now, the expression is \(0.25 + 3(-1)^{3}\).

Step 3: Evaluate the Exponent

Calculate \((-1)^{3}\), which equals \(-1\). Thus, the expression becomes \(0.25 + 3(-1)\).

Step 4: Multiply Coefficient

Multiply \(3\) by \(-1\), which results in \(-3\). Now, the expression is \(0.25 - 3\).

Step 5: Perform the Addition

Finally, calculate \(0.25 - 3\), resulting in \(-2.75\).

Key Concepts

Substitution

Substitution is the process of replacing variables with their assigned values. When you see an algebraic expression with variables, the first step is usually to substitute. In our example, the expression given is \(b + 3(a+d)^{3}\). Here, each letter represents a numerical value:

• \(a = -5\)
• \(b = 0.25\)
• \(d = 4\)

To substitute, simply replace each variable with the corresponding value. The expression \(b + 3(a+d)^{3}\) then transforms into \(0.25 + 3(-5 + 4)^{3}\). Substitution is crucial because it allows us to work with numbers instead of variables, making the expression much easier to handle.

Order of Operations

The order of operations is a set of rules that determines the sequence in which operations should be performed in a mathematical expression. Performing operations in the correct order is essential to ensure accurate results. The universally accepted order is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

In the expression \(0.25 + 3(-5 + 4)^{3}\), we start by simplifying inside the parentheses, finding \((-5 + 4)\), which gives \(-1\). We then move on to the exponent, \((-1)^{3}\), before handling multiplication and finally addition or subtraction. Adhering to the order of operations avoids common mistakes that can lead to incorrect solutions.

Exponents

Understanding how to work with exponents is another important concept in algebra. Exponents indicate repeated multiplication of a number by itself. The expression \((a+d)^{3}\) involves raising a number to the power of 3, which means multiplying the number by itself three times. In our solved example, we have \((-1)^{3}\).To calculate this:

• First multiply \(-1\) by itself: \(-1 \times -1 = 1\).
• Then multiply that result by \(-1\): \(1 \times -1 = -1\).

So, \((-1)^{3} = -1\). This step is pivotal because it changes the direction of computation for the rest of the expression.

Simplification

Simplification is the process of making expressions easier to work with by combining like terms and performing arithmetic operations. After substitution and following the order of operations, we face an expression like \(0.25 + 3(-1)\). Here, it is important to:

• Multiply 3 by \(-1\), which results in \(-3\).
• Add \(0.25\) to \(-3\), which gives \(-2.75\).

Simplification can involve several different kinds of steps like combining terms or reducing fractions, but ultimately, each step should make the expression clearer and closer to a final answer. In this case, the final simplified answer is \(-2.75\). Mastering simplification ensures that our mathematical communication is concise and clear.