Problem 84 - Elementary Algebra Within Reach

Question

Evaluate the expression. $$(-5)^{3}+3$$

Step-by-Step Solution

Answer

The expression \( (-5)^{3}+3 \) evaluates to -122.

Step 1: Calculating Power

First calculate \( (-5)^{3} \). Because the base is -5 and the exponent is 3, and since the exponent is odd, the result will be -125.

Step 2: Applying Addition

Now add 3 to the result: \( -125 + 3 \) which gives -122.

Key Concepts

Evaluate Expressions

When you evaluate an expression, you're finding out what value it represents. Think of an expression as a mini math sentence that you need to solve. The key steps are to follow the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right)).
For example, in the expression \((-5)^3 + 3\), you focus on solving the exponent first before tackling the addition. This structured approach ensures that you always arrive at the correct solution, so remember to stay organized and tackle one operation at a time.

Powers of Negative Numbers

Dealing with powers of negative numbers can be tricky, but it's not so bad once you get the hang of it. When you raise a negative number to a power, the outcome depends on whether the exponent is odd or even.

• If the exponent is odd, like in \((-5)^3\), the result is negative. This is because the negative sign gets repeated an odd number of times, ending up as negative.
• If the exponent is even, the result is positive because the negative signs pair up and cancel each other out. For example, \((-5)^2 = 25\).

Always keep this principle in mind: odd exponents maintain a negative result, while even exponents flip it to positive.

Additive Operations

Once you have calculated the power of the negative number, the next step is to handle the addition part of the expression. Additive operations are one of the most basic math operations, but they are essential for solving more complex equations.
In the expression \((-5)^3 + 3\), after determining that \((-5)^3 = -125\), the next step is to add 3 to -125.

• Start by imagining a number line. You're standing at -125 and moving 3 steps to the right. Where do you land?
• Addition can be thought of as moving in this direction along the number line, which helps visualize and perform the operation correctly.

This results in -122. Understanding that addition essentially means counting forward (or backward if subtracting) allows you to solve even large and complex expressions with ease.