Problem 82 - Prealgebra

Question

Given \(\mathrm{a}=4.1, \mathrm{~b}=-1.8\), and \(\mathrm{c}=-9.5\), evaluate the expression \(\mathrm{a}-\mathrm{bc}^{2}\).

Step-by-Step Solution

Answer

The expression evaluates to 166.55.

Step 1: Substitute the values into the expression

Start by replacing each variable in the expression with its corresponding given value. The expression becomes:\[ 4.1 - (-1.8) imes (-9.5)^2 \]

Step 2: Calculate the power of the expression

Evaluate the power in the expression. Calculate \((-9.5)^2\), which is \[ (-9.5) imes (-9.5) = 90.25 \]

Step 3: Multiply the results

Next, we multiply the results of the power operation by \(-1.8\):\[ -1.8 imes 90.25 = -162.45 \]

Step 4: Final subtraction

Subtract the result of the multiplication from 4.1:\[ 4.1 - (-162.45) = 4.1 + 162.45 = 166.55 \]

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with their given numerical values. This is crucial for simplifying and evaluating expressions effectively. By substituting, you can transform a general expression into something specific that can be calculated.
For example, in the expression \( a - bc^2 \), if \( a = 4.1 \), \( b = -1.8 \), and \( c = -9.5 \), you replace \( a, b, \) and \( c \) with these values. This helps in arriving at:

• \( 4.1 - (-1.8) \times (-9.5)^2 \)

Substitution allows you to simplify the problem and prepares you for further calculations. This step is essential to accurately perform computations as per the numerical values provided.

Order of Operations

Order of Operations is a fundamental concept in mathematics used to determine the sequence in which operations should be performed to correctly evaluate an expression. The standard order of operations is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our example, applying the order of operations is key. Let's look at the expression \( 4.1 - (-1.8) \times (-9.5)^2 \):

• Handle the exponent first: \((-9.5)^2\)
• Next, perform the multiplication: \((-1.8) \times 90.25\)
• Finally, do the subtraction: \(4.1 - (-162.45)\)

By following these steps in the prescribed order, you ensure the calculations are accurate and maintain mathematical integrity. Deviating from the order of operations will lead to incorrect results.

Exponents

Exponents are used in mathematics to denote repeated multiplication of a number by itself. The expression \(c^2\) is read as "\(c\) squared" and implies \(c\times c\).
In our example, evaluating \((-9.5)^2\) means you multiply -9.5 by itself:

• First calculation: \((-9.5) \times (-9.5)\)
• Result: 90.25

When applying exponents, it’s important to pay attention to the signs involved. A negative number squared results in a positive number because multiplying two negative numbers gives a positive result. This is key in simplifying expressions that involve powers.

Negative Numbers

Understanding negative numbers is essential when performing calculations involving such values. Negative numbers are values less than zero and show the opposite of positive numbers.
In our exercise, both \(b\) and \(c\) are negative, and it affects how we calculate expressions like the multiplication and exponentiation:

• The square of a negative \((-9.5)^2\) becomes positive (90.25). This happens because multiplying two negatives results in a positive.
• Multiplying the positive result (90.25) by a negative number (-1.8) results in a negative product (-162.45).

Additionally, remember that subtracting a negative is equivalent to adding its positive counterpart. Hence, \(4.1 - (-162.45)\) simplifies to \(4.1 + 162.45\).
Handling negative numbers properly ensures accurate mathematical operations and correct results.