Problem 86 - Prealgebra

Question

Given \(\mathrm{a}=7.45, \mathrm{~b}=-6.1\), and \(\mathrm{c}=-3.5\), evaluate the expression \(\mathrm{a}-\mathrm{bc}^{2}\).

Step-by-Step Solution

Answer

The value of the expression is 82.175.

Step 1: Identify the Expression to Evaluate

The given expression is \( a - bc^2 \). We are to evaluate it using the provided values for \( a \), \( b \), and \( c \).

Step 2: Substitute Given Values

Substitute \( a = 7.45 \), \( b = -6.1 \), and \( c = -3.5 \) into the expression \( a - bc^2 \). It becomes \( 7.45 - (-6.1)(-3.5)^2 \).

Step 3: Calculate \( c^2 \)

Calculate \( c^2 \) where \( c = -3.5 \). So, \( c^2 = (-3.5)^2 = 12.25 \).

Step 4: Compute \( bc^2 \)

Calculate \( bc^2 = (-6.1) imes 12.25 \). Multiply \(-6.1\) by \(12.25\) to get \(-74.725\).

Step 5: Evaluate the Final Expression

Substitute \( bc^2 = -74.725 \) back into the expression \( a - bc^2 \), which becomes \( 7.45 - (-74.725) \). Simplify it to \( 7.45 + 74.725 = 82.175 \).

Key Concepts

Substitution in Algebraic Expressions

Substitution involves replacing the variables in an algebraic expression with their given numerical values. When an exercise provides specific values for variables, like in this example, you simply substitute each occurrence of the variable with its respective value. For example, in the expression \( a - bc^2 \), we replace \( a \) with 7.45, \( b \) with -6.1, and \( c \) with -3.5. This is how we transform an abstract expression into a computational one: \( 7.45 - (-6.1)(-3.5)^2 \). Steps for Substitution:

• Identify each variable in the expression.
• Find the given value for each variable.
• Replace the variable in the expression with its corresponding value.

By using substitution, you turn potentially confusing symbols into numbers you can work with directly.

Understanding the Order of Operations

Evaluating expressions requires a systematic approach provided by the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates how we proceed with calculations to reach the correct answer.- **Parentheses:** Solve operations inside parentheses first. In the expression \( 7.45 - (-6.1)(-3.5)^2 \), there are no parentheses, but understanding this step is crucial.- **Exponents:** Calculate any exponents before moving forward. Here, \( (-3.5)^2 \) is solved first, resulting in 12.25.- **Multiplication and Division:** Next, handle multiplication or division from left to right. The multiplication \( (-6.1) \times 12.25 \) is solved, leading to \(-74.725\).- **Addition and Subtraction:** Finally, perform any addition or subtraction. Subtract \(-74.725\) from 7.45, which simplifies to adding 74.725, yielding a result of 82.175.This systematic approach ensures accuracy and precision in solving mathematical expressions.

Arithmetic Operations Simplified

Arithmetic operations include the fundamental processes of addition, subtraction, multiplication, and division. These operations are the backbone of evaluating expressions.1. **Addition and Subtraction:** In evaluating \( a - bc^2 \), the subtraction \( a - bc^2 \) transforms into addition when subtracting negative values, leading to 7.45 + 74.725. Understanding that subtracting a negative is equivalent to addition is essential.2. **Multiplication:** Multiplying numbers involves combining equal groups. In \( (-6.1) \times 12.25 \), you compute the product to be \(-74.725\). Multiplication with negative numbers alters the sign of the result.3. **Exponents:** Raises a number to a specific power. In this task, \( (-3.5)^2 \) computes the base number, -3.5, multiplied by itself to give 12.25.By mastering these arithmetic rules and operations, we can comprehend expressions more effectively, solving them accurately step-by-step.