Problem 90
Question
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
Verified Answer
The numerical expression for the sentence is 25.
1Step 1: Formulate the Mathematical Expression
The first part of the sentence 'Cube -5.' tells to cube the number -5, which is written mathematically as \((-5)^3\). The second part of the sentence 'Subtract this exponential expression from -100.' tells to subtract the result obtained from the previous expression from -100. The mathematical expression for the whole sentence will be \(-100 - (-5)^3\).
2Step 2: Calculate Cube of -5
The cube of -5 is calculated as \((-5)^3 = -125\). So, the expression becomes \(-100 - (-125)\).
3Step 3: Final Simplification
Subtracting a negative number is equivalent to adding its absolute value. So, \(-100 - (-125) = -100 + 125 = 25\).
Key Concepts
Numerical ExpressionExponentiationSubtractionNegative Numbers
Numerical Expression
In mathematics, a numerical expression is a combination of numbers and operations that helps us to solve problems involving calculations. It usually includes operators like addition, subtraction, multiplication, and division, and sometimes also includes numbers with exponents. To simplify a numerical expression means to perform the operations in a sequence, governed by specific rules known as the order of operations.
The process of creating a numerical expression starts with understanding the verbal sentence describing the problem. In this case, we translate the sentence 'Cube -5 and subtract this from -100.' into a numerical expression. This gives us \[ -100 - (-5)^3 \].
Once the expression is set, you move on to simplification by following mathematical rules. This enables you to find a final answer and solve the problem accurately.
The process of creating a numerical expression starts with understanding the verbal sentence describing the problem. In this case, we translate the sentence 'Cube -5 and subtract this from -100.' into a numerical expression. This gives us \[ -100 - (-5)^3 \].
Once the expression is set, you move on to simplification by following mathematical rules. This enables you to find a final answer and solve the problem accurately.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It implies repeated multiplication of the base by itself. For example, cubing \( -5 \) is a form of exponentiation, where \( -5 \) is multiplied by itself twice more: \[ (-5) \times (-5) \times (-5) \].
The result of cubing \(-5\) is \(-125\). This operation is crucial in the problem as it changes the original number significantly. When dealing with a negative base like \(-5\), remember that the signs also multiply through the process:
This is why \((-5) \times (-5) = 25\), but \(25 \times (-5) = -125\). It’s key to handle the signs carefully in exponentiation.
The result of cubing \(-5\) is \(-125\). This operation is crucial in the problem as it changes the original number significantly. When dealing with a negative base like \(-5\), remember that the signs also multiply through the process:
- Negative times Negative = Positive
- Positive times Negative = Negative
This is why \((-5) \times (-5) = 25\), but \(25 \times (-5) = -125\). It’s key to handle the signs carefully in exponentiation.
Subtraction
Subtraction is one of the basic arithmetic operations where you take away one number from another. In mathematical expressions, it is denoted by the symbol \(-\). Subtraction involves determining the difference between two numbers, which is essential in solving many problems.
In the problem, we have the expression \(-100 - (-125)\). When subtracting a number that has a subtraction sign in front, it involves a certain rule in mathematics:
In the problem, we have the expression \(-100 - (-125)\). When subtracting a number that has a subtraction sign in front, it involves a certain rule in mathematics:
- Subtracting a negative number is equivalent to adding its opposite.
Negative Numbers
Negative numbers are numbers that are less than zero, often represented with a minus sign \(-\). They are essential in both real-world contexts and theoretical mathematics for representing quantities below a baseline value, such as temperatures below zero.
When working with negative numbers, especially in operations like multiplication and exponentiation, it’s crucial to remember the rules of signs. In the expression we evaluated, handling of negative numbers occurs both in the base for cubing, \((-5)^3\), and in subtraction steps, \(-100 - (-125)\).
Dealing with negative numbers can be confusing at first, but consistent application of basic rules helps in working through them efficiently. In practical problems, they often represent deficits, debts, or losses.
When working with negative numbers, especially in operations like multiplication and exponentiation, it’s crucial to remember the rules of signs. In the expression we evaluated, handling of negative numbers occurs both in the base for cubing, \((-5)^3\), and in subtraction steps, \(-100 - (-125)\).
- A negative number multiplied by another negative results in a positive.
- A negative cubed or multiplied by a positive remains negative.
Dealing with negative numbers can be confusing at first, but consistent application of basic rules helps in working through them efficiently. In practical problems, they often represent deficits, debts, or losses.
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Problem 89
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