Problem 49 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ (-7.9)^{2} $$

Step-by-Step Solution

Answer

The result of \\((-7.9)^2\\) is \\62.41\\.

Step 1: Understanding the Expression

The expression \((-7.9)^2\) is asking us to find the square of \(-7.9\). This means we will multiply \(-7.9\) by itself.

Step 2: Setting Up the Multiplication

To evaluate the expression, write it as a multiplication problem: \(-7.9 \times -7.9\).

Step 3: Performing the Multiplication

Multiply \7.9 \times 7.9\. This yields \62.41\ (since \7.9 \times 7.9 = 62.41\). As negative times negative is positive, \(-7.9) \times (-7.9) = 62.41\.

Step 4: Final Answer

Confirm the multiplication result for accuracy. Thus, the final result of \(-7.9)^{2}\) is \62.41\.

Key Concepts

Understanding Exponents

Exponents are a mathematical way to express repeated multiplication of the same number. When you see a number like \((-7.9)^2\), it indicates that the number \(-7.9\) should be multiplied by itself. The small floating number 2 is called the exponent and tells us how many times the base number is used as a factor. In this case, you multiply \(-7.9\) by \(-7.9\) once. Exponents make it easier to write and understand large calculations without writing all of the multiplication steps. This notation is a shorthand that helps in simplifying expressions and solving problems more efficiently.

Handling Negative Numbers

Negative numbers can sometimes be tricky, especially when they involve exponents. A negative sign in front of a number denotes its direction on the number line; for example, \(-7.9\) is 7.9 units left of zero. When you square a negative number, you are multiplying the number by itself, which may seem confusing at first. However, the rule is straightforward:

• Negative times negative equals positive.
• Positive times positive equals positive.

Therefore, when you square \(-7.9\), you are essentially multiplying two negative numbers, resulting in a positive value. In this exercise, \(-7.9) \times (-7.9) = 62.41\). Remember, squaring a negative number always yields a positive result.

The Power of Multiplication

Multiplication is one of the fundamental operations in mathematics. It involves combining equal groups of objects or numbers. When you square a number, you use multiplication to calculate its square. The expression \((-7.9)^2\) was broken down into a multiplication problem: \(-7.9 \times -7.9\). Here's why multiplication is essential:

• It helps in finding areas and volumes.
• It simplifies complex calculations.

In our example, we first simplified the problem to understand that we multiply \(-7.9\) times itself. This step-by-step approach ensures accuracy and eases the complexity of solving such problems.

Algebra: Bringing It All Together

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It's a tool that connects the ideas of numbers, basic operations, and real-world problems. The expression \((-7.9)^2\) illustrates an algebraic operation: using exponents to denote multiplication of a negative number by itself.Key points to consider when dealing with algebraic expressions include:

• Understanding and applying mathematical operations like squaring.
• Simplifying expressions to reveal meaningful results.

Algebra helps us move from real-world situations to symbolic representations, making it easier to analyze and find solutions to complex problems. This particular exercise is an excellent example of how algebra transforms a straightforward expression into a meaningful calculation that is easy to solve.