Problem 89 - Introductory Algebra for College Students

Question

Solve each equation .Use a calculator to help with the arithmetic. Check your solution using the calculator. Evaluate: \(-10^{2}\).

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

Firstly, understand the expression \(-10^{2}\). This represents negative of ten raised to the power of two. The order of operations must be applied. BIDMAS dictates that the exponentiation (Indices) is performed before the negation (which is equivalent to multiplication by -1)

Step 2: Apply the Exponent

Apply the exponent to the base which is 10. \(10^{2} = 10 * 10 = 100\).

Step 3: Apply the Negation

Finally, apply negation to the result obtained from the previous step. The negative sign changes the sign of 100 to -100.

Key Concepts

Exponents

Exponents are a powerful tool in algebra, allowing us to express repeated multiplication concisely. When you see a number raised to a power, such as in \(10^2\), the number 10 is the base and 2 is the exponent. This notation means that you multiply the base by itself as many times as indicated by the exponent. Therefore, \(10^2\) means \(10 \times 10\), which equals 100.

Exponents can greatly simplify arithmetic involving large numbers. Instead of multiplying 10 by itself repeatedly, exponents allow us to write this as a simple expression. It’s important to clearly understand the relationship between the base and the exponent to avoid common mistakes. For example, don't confuse \(10^2\) with \(2^{10}\), which have very different values.

• When the exponent is 2, it's also known as "squaring" a number.
• Exponents are not limited to just whole numbers; they can be fractions, negatives, or even variables.

Order of Operations

When solving equations involving multiple operations, such as exponents and negations, it's crucial to follow the correct order to ensure accurate results. The standard rule to remember is BIDMAS (or PEMDAS in some places), standing for: Brackets, Indices (or Exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

In the expression \(-10^{2}\), applying BIDMAS tells us to handle the exponent before dealing with the negative sign. Here’s the breakdown:

• First, evaluate the exponent: \(10^2 = 100\).
• Next, apply the negative sign, which is like multiplying by -1: \(-1 \times 100 = -100\).

Following the right order of operations is key to arriving at the correct answers in algebra problems. Incorrect application can easily lead to wrong conclusions, especially when negations and powers are involved.

Negative Numbers

Negative numbers are essential in mathematics, representing values less than zero. Understanding how they interact with different operations is fundamental when dealing with algebraic expressions. The initial expression \(-10^{2}\) utilizes negative numbers to indicate that the entire resulting value will be negative.

Here are some key points to remember when working with negative numbers:

• When multiplying or dividing two numbers with the same sign, the result is positive. However, with different signs, the result is negative.
• Adding a negative number is the same as subtracting its positive counterpart (e.g., \(10 + (-5) = 10 - 5\)).
• The negative of a negative number is positive (\(-(-x) = x\)).

Thus, understanding these foundational rules helps simplify and solve algebraic expressions involving negative numbers, ensuring that calculations, especially with exponents, yield correct results.