Problem 89 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate: \(-10^{2}\)

Step-by-Step Solution

Answer

The evaluation of \(-10^2\) is -100.

Step 1: Interpret the expression

The expression can be interpreted as -1 multiplied by 10 to the power of 2.

Step 2: Apply the exponent

10 squared is 100. So now we have \(-1 * 100\).

Step 3: Apply multiplication

-1 times 100 equals -100.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, writing \(10^2\) is just a shorthand for saying 10 multiplied by itself once, which gives us:

• 10 * 10 = 100

So, when you see 10 raised to the power of 2, you can think of it as 10 times 10, resulting in 100. Exponents help simplify expressions, making them easier to manage especially when dealing with very large or very small numbers. It's essential to understand that the exponent tells us how many times to use the base (in this case, 10) in a multiplication operation.
When encountering negative numbers, placing a negative sign outside an expression with an exponent, as in \(-10^2\), doesn't change what's inside the exponent; it only affects the final result after calculation.

Navigating the Order of Operations

The order of operations is a standard rule we follow to ensure mathematical expressions are evaluated correctly. A popular way to remember the order is the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (ie. powers and square roots, etc.)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

When you look at \(-10^2\), it's crucial to respect these rules. First, address any exponents. Here, 10 squared is done first, giving us 100. Then, once the exponents are resolved, apply other operations like multiplication or division, in this case, multiplying by -1.
This rule ensures we all interpret expressions in the same way, avoiding confusion and errors that could arise from calculating in different sequences.

Executing Multiplication with Negative Numbers

Multiplication of numbers, especially with negatives, is pivotal in mathematics. A negative times a positive results in a negative product. For instance, when evaluating \(-1 * 100\), the operation involves multiplying 1 by 100 to get 100 and then attaching the negative sign:

• -1 * 100 = -100

It's important to remember:

• A positive times a positive is positive
• A negative times a positive is negative
• A negative times a negative is positive

This consistent set of rules helps provide clarity and accuracy in calculations. So, in the context of \(-10^2\), the multiplication step plays a key role in delivering the final answer of -100.