Problem 29 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$-5^{2}-4^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-41\).

Step 1: Understand the Expression

The expression given is \(-5^{2} - 4^{2}\). It's important to recognize that the negative sign is outside the exponent, so it stays as negative until the last operations.

Step 2: Calculate 5 Squared

Calculate \(5^{2}\): \(5^{2} = 25\)

Step 3: Apply the Negative Sign to 5 Squared

Apply the negative sign to the result from Step 2: \( -5^{2} = -25 \)

Step 4: Calculate 4 Squared

Calculate \(4^{2}\): \(4^{2} = 16\)

Step 5: Subtract 4 Squared from Negative 5 Squared

Subtract the value of \(4^{2}\) from \(-5^{2}\): \(-25 - 16 = -41\)

Step 6: Final Simplified Result

The expression \(-5^{2} - 4^{2}\) simplifies to \(-41\).

Key Concepts

Order of Operations

When simplifying mathematical expressions, following the order of operations is crucial to arriving at the correct answer. Here's a quick way to remember the order:

• Parentheses
• Exponents (including squared numbers)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

If you ever heard of the acronym PEMDAS or "Please Excuse My Dear Aunt Sally," it stands for these steps and helps remember the correct order. For example, in the problem \(-5^{2} - 4^{2}\), we have to perform exponentiation (which is squaring 5 and 4) before other operations like subtraction. Always handle operations like exponents first to ensure an accurate outcome.

Squared Numbers

Squared numbers are numbers that are multiplied by themselves. When you see an expression like \(5^{2}\), it means \(5\times 5\). So, in this specific exercise, \(5^{2}\) equals 25. Similarly, \(4^{2}\) means \(4\times 4\), which gives us 16.

Working with squared numbers helps simplify expressions efficiently, but it’s important to apply any outside negative signs only after squaring the numbers. For example, if you have \(-5^2\), you should first calculate \(5^2 = 25\), and then apply the negative sign to get \(-25\). It's a common error to combine the negative sign with the number before applying the exponent, which could lead to incorrect results.

Negative Numbers

Negative numbers can sometimes be tricky when they appear in expressions, especially near exponents. A key point here is to apply the negative sign last when dealing with exponents, unless parentheses indicate otherwise. In our exercise, the computation proceeds as:

• Calculate the square, as normal.
• Apply the negative sign if it's not enclosed in parentheses.

So for \(-5^{2}\), you first process \(5^2\) to get 25 and then apply the negative, turning it into \(-25\). Remember that the placement of parentheses greatly impacts how negative numbers are handled. If the expression had been \((-5)^2\), both the negative and the 5 would be squared, resulting in a different value. When simplifying expressions with negative numbers, careful attention is needed to avoid mistakes.