Problem 77 - Prealgebra

Question

Simplify the given expression. \((6.3)(1.88)-(-2.2)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 7.004.

Step 1: Understand the Expression

First, examine the expression to understand the components involved: \((6.3)(1.88)-(-2.2)^{2}\). This expression involves multiplication in the first term and a power in the second term.

Step 2: Calculate the Multiplication

Perform the multiplication for the first term:\(6.3 \times 1.88\). Calculate it step by step or using a calculator:\(6.3 \times 1.88 = 11.844\).

Step 3: Calculate the Square

Calculate the square of \(-2.2\):\((-2.2)^{2}\).Remember that squaring a negative number results in a positive value:\((-2.2) \times (-2.2) = 4.84\).

Step 4: Substitute and Simplify

Substitute the results from Steps 2 and 3 back into the expression:\(11.844 - 4.84\).Subtract the values to simplify:\(11.844 - 4.84 = 7.004\).

Key Concepts

Understanding Multiplication

Multiplication is one of the simplest arithmetic operations, yet it's foundational for mathematics. In our expression, we encounter multiplication in (6.3)(1.88). To simplify multiplication, think of it as repeated addition. For instance, multiplying 6.3 by 1.88 means adding 6.3 to itself 1.88 times.

To confidently perform multiplication involving decimals, it often helps to:

• Ignore the decimals temporarily and multiply the numbers as if they're whole numbers.
• Count the total number of decimal places in the factors and ensure the product has the same number of decimal places.

Let's break it down: 6.3 and 1.88 have a total of three decimal places. After multiplying as whole numbers, insert the decimal back to match the original places, ensuring 11.844 is the accurate result.

Exponents and Their Role

Exponents can sometimes be tricky, but they are a simple concept when broken into steps. They represent repeated multiplication. For example, (-2.2)^{2} means multiplying -2.2 by itself once.

• For positive bases, raising a number to an exponent like 2 (or squaring it) always increases the size.
• With negative bases like -2.2, the negative sign plays a key role. However, squaring the number means the negatives cancel each other out, resulting in a positive product.

As with our given calculation, ((-2.2) imes (-2.2)) becomes 4.84. Remember, any even exponent of a negative number will yield a positive number.

Navigating Negative Numbers

Negative numbers, though sometimes daunting, follow consistent rules. In our problem, they appear within the task of finding (-2.2)^{2}. Let's demystify them:

Using negative numbers in multiplication or squaring follows specific rules:

• Multiplying a positive by a negative results in a negative.
• Multiplying a negative by a negative results in a positive.

When you square any negative number, the negative signs effectively cancel each other out due to multiplication rules, resulting in a positive outcome. So, you're left with 4.84 after calculating (-2.2)^2. These consistent rules make handling negative numbers predictable and manageable.