Problem 90 - Introductory Algebra for College Students

Question

In Exercises \(89-92,\) express each sentence as a single numerical expression. Then use the order of operations to simplify the expression. Cube \(-5 .\) Subtract this exponential expression from \(-100 .\)

Step-by-Step Solution

Answer

The result of the given operations is \(25\).

Step 1: Cube the number -5

This operation requires the multiplication of -5 by itself two more times. That is, evaluating \((-5)^3\). This results in \(-125\) because a negative number multiplied by another negative number gives a positive number, but then multiplied again by a negative number turns back to being negative.

Step 2: Subtract the result from -100

The second part of the exercise instructs us to subtract the result from \(-100\). This means we perform the operation \(-100 - (-125)\). Remember that subtracting a negative is the same as adding a positive.

Step 3: Solve the subtraction

Apply the rule of subtracting a negative equals adding a positive gives us \(-100 + 125\). The result of this operation is \(25\).

Key Concepts

Cubing Numbers

Cubing numbers refers to raising a number to the third power. This means multiplying the number by itself three times. For example, in our exercise, we were asked to cube \(-5\).

• Firstly, multiply \(-5 \, \times \, -5\) to get \(25\). Multiplying two negative numbers results in a positive product.
• Next, multiply the result by \(-5\) again: \(25 \, \times \, -5 = -125\). This time, multiplying a positive number by a negative number results in a negative product.
• The cube of a number gives us an idea about its volumetric measure, as it often represents the space within a three-dimensional object with sides of that length.

Cubing is a crucial mathematical operation following its placement in the order of operations. It alters a number significantly, especially when dealing with very large or very small base numbers.

Negative Numbers in Algebra

Negative numbers often seem tricky, but with a few rules, they can become easy to manage. When dealing with negative numbers, the sign can hugely impact calculations.

• Multiplication and Division: Multiplying or dividing two negative numbers gives a positive result. However, if a positive and a negative number are involved, the outcome is negative.
• Order Matters: The order in which you apply operations can affect the sign. Always keep the order of operations in mind: parentheses, exponents, multiplication/division (left to right), addition/subtraction (left to right).

In our exercise, the key was recognizing that subtracting a negative is equivalent to adding a positive. So, \(-100 - (-125)\) becomes \(-100 + 125\). This crucial step shows how understanding the rules for negative numbers can simplify complex-looking algebraic expressions.

Numerical Expressions

Numerical expressions are combinations of numbers and operations that need to be simplified or solved. They often involve different mathematical operators, such as addition, subtraction, multiplication, division, and exponents.

• Expression Formation: In our exercise, we first expressed the statement "Cube \(-5\) and subtract from \(-100\)" as a numerical expression: \(-100 - (-5)^3\).
• Simplification: By following the correct order of operations (often remembered using "PEMDAS"—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we simplify the expression to its simplest form, ultimately giving us \(25\).

Being comfortable with creating and simplifying numerical expressions is critical for solving various mathematical problems. It helps transition real-world situations into mathematical language, allowing for clear, systematic problem solving.