Problem 40 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (-2)^{2}-3(-2)(6)-(-5)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is -57.

Step 1: Evaluate the Squared Terms

First, evaluate \((-2)^{2}\) and \((-5)^{2}\). This results in \((-2)^{2} = 4\) and \((-5)^{2} = 25\) because squaring a negative number results in a positive number.

Step 2: Simplify the Middle Term

Now, calculate the middle term \(-3(-2)(6)\). This results in \(-3 \times (-2) \times 6 = 36\). Multiplying two negative numbers results in a positive number, so the middle term simplifies to +36.

Step 3: Substitute the Simplified Values Back into the Expression

Substitute the evaluated terms back into the original expression:\( 4 - 36 - 25 \).

Step 4: Simplify the Expression

Start by simplifying \(4 - 36 = -32\), then simplify \(-32 - 25 = -57\). Therefore, the simplified expression is \(-57\).

Key Concepts

Evaluating Powers

When you come across expressions with exponents, like \((-2)^2\) or \((-5)^2\), the process is known as evaluating powers. In mathematics, the *power* of a number tells you how many times to use the number in a multiplication. For example, \((x)^3\) means to multiply \(x\) by itself three times: \((x \times x \times x)\).

Remember, when dealing with negative numbers, squaring them always results in a positive number. This is because multiplying two negative numbers gives a positive result. So, \((-2)^2\) becomes \(4\). Similarly, \((-5)^2\) becomes \(25\).

Here’s a quick tip to identify mistakes with powers:

• Always check the sign of the base number. If it’s negative, squaring it should result in a positive.
• Make sure to apply the exponent to the entire base number, especially if the base includes a negative sign.

Multiplying Negative Numbers

When it comes to multiplying negative numbers, there’s a simple rule that says multiplying two negative numbers gives a positive number. But let’s break it down further to understand why this happens.

When numbers are negative, think of them as being in the opposite direction on the number line. So, multiplying two negative numbers is akin to changing direction twice, which brings you back to positive territory.

In the example \(-3(-2)(6)\), you might do it step by step:

• First, multiply \(-3\) and \(-2\). This gives \(6\) because turning twice from negative goes positive.
• Then multiply \(6\) by \(6\), resulting in \(36\).

Keep in mind the key rules for multiplication:

• Negative times negative is positive.
• Negative times positive remains negative.

Apply these rules carefully and your calculations will be spot-on.

Order of Operations

The order of operations can often seem tricky, but it’s an essential concept to understand for simplifying expressions correctly. Mathematicians agree to follow a specific sequence to ensure consistency in calculations. This sequence is known as PEMDAS or BIDMAS, which stands for:

• Parentheses or Brackets first.
• Exponents or Indices (such as squaring).
• Multiplication and Division (from left to right).
• Addition and Subtraction (from left to right).

In the given expression \( (-2)^{2}-3(-2)(6)-(-5)^{2} \), you first evaluated the powers (exponents) to \(4\) and \(25\). Then, you multiplied the terms involving negative and positive numbers. Finally, you simplified the expression from left to right starting with subtraction operations.

Following the correct order ensures that every mathematical problem is simplified the same way across different contexts, providing clear, reliable results. Remember to take your time and process each part of an expression step-by-step.