Problem 118 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(a b,\) when \(a=452\) and \(b=-0.86\)

Step-by-Step Solution

Answer

The value of the expression \(a b\) for \(a=452\) and \(b=-0.86\) is -388.72.

Step 1: Substitute the values into the expression

Replace \(a\) with 452 and \(b\) with -0.86 in the expression \(a b\). This gives us \(452*(-0.86)\).

Step 2: Carry out the multiplications

Perform the multiplication which, when rounded to two decimal places, gives -388.72.

Key Concepts

Understanding Substitution

Substitution is a method used to replace variables in an expression with their actual numeric values. In the given exercise, you have been asked to evaluate the expression \(a \times b\) for specific values of \(a\) and \(b\).

Literally, the word "substitute" means "to put in place of." Here, you substitute \(a\) by 452 and \(b\) by -0.86. This transforms the expression \(a \times b\) into \(452 \times (-0.86)\). By correctly substituting the variables, you set the stage for solving the expression accurately.

• Replace each variable with its corresponding value.
• Ensure calculations are done using the substituted values.

Substitution is an essential step in algebra, simplifying expressions for further operations, like multiplication in this case.

The Process of Multiplication

Multiplication involves combining a number multiple times, as specified by the multiplier. When tasked with multiplying two numbers, as in the expression \(452 \times (-0.86)\), you are essentially adding 452 to itself, -0.86 times.

This operation is straightforward when dealing with positive numbers, but when negative numbers come into play, it affects the final sign of the product. The rules of multiplication state that a positive number times a negative number results in a negative product, which is why the answer is negative in this case.

• Identify the numbers to be multiplied.
• Apply multiplication rules for positive and negative numbers.
• Perform calculations accurately.

Multiplying precisely, especially in areas involving positive and negative numbers, ensures the accuracy of your result.

Role of Variables

Variables are symbols used to hold values that can vary. In expressions, such as \(a \times b\), \(a\) and \(b\) are variables representing numbers. They provide flexibility and generality, allowing the same expression to be used with different numbers.

In the exercise, the variables \(a\) and \(b\) were placeholders for 452 and -0.86, which were substituted in place.

• Variables act as placeholders for numbers.
• They allow expressions to be flexible and reusable.
• Once replaced with values, variables transform the expression into a solvable numerical equation.

Understanding variables' role helps in manipulating and solving algebraic expressions effortlessly.

Impact of Negative Numbers

Negative numbers have unique properties that result in a change in sign when used in mathematical operations. In multiplication, if one of the numbers is negative, the result becomes negative, as seen in the expression \(452 \times (-0.86) = -388.72\).

This outcome follows basic multiplication rules: multiplying a positive by a negative yields a negative product. It's crucial to remember this rule when dealing with expressions involving negative numbers.

• Be aware of the sign of numbers involved in operations.
• Understand that negative numbers change the sign of the product.
• Apply rules consistently for accurate calculations.

Grasping the concept of negative numbers is vital for performing arithmetic operations correctly and interpreting results accurately.