Problem 100 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ -x^{3} \text { for } x=5 $$

Step-by-Step Solution

Answer

The value of the expression is -125.

Step 1: Substitute the value of the variable

In the expression \[ -x^3 \]replace \(x\) with the given value \(5\). This gives us:\[ -(5)^3 \].

Step 2: Calculate the power

Calculate \[ 5^3 \] which means multiplying the number 5 by itself three times: \[ 5 \times 5 \times 5 = 125 \].

Step 3: Apply the negative sign

Now apply the negative sign, so \[ -125 \] becomes the result of the expression evaluated for \( x = 5 \).

Key Concepts

Power of a Number

In algebra, the power of a number refers to how many times a number is multiplied by itself. This is also known as exponentiation. When you see something like

• \(5^3\), it means you multiply 5 by itself twice more: \(5 \times 5 \times 5\).
• The number 5 is called the base, and the number 3 is called the exponent.

Breaking it down, you can think of the exponent as instructions for repeated multiplication. The process involves multiplying the base by itself a number of times specified by the exponent.
For example:

• \(5^2 = 5 \times 5 = 25\)
• \(5^3 = 5 \times 5 \times 5 = 125\)

Therefore, knowing how to work with powers is crucial in algebra, especially when solving expressions involving exponentiated variables. By mastering this, you'll have an easier time evaluating expressions in various algebraic contexts.

Substitution in Algebra

In algebra, substitution is a fundamental process used to evaluate expressions. This involves replacing a variable with a specific number or another expression. For example, if you have the expression -\(x^3\) and you're given that \(x = 5\), you'd substitute the 5 wherever the \(x\) appears. This gives us a new expression to work with:

• -\((5)^3\)

The goal of substitution is to simplify the expression to a numerical value. This is done by conducting operations such as addition, multiplication, or exponentiation with the substituted values. Practicing this skill means you can turn a general expression into a specific number, allowing you to solve equations and analyze expressions effectively. While this might sound straightforward, be mindful of operator precedence and order of operations to arrive at the correct solution.

Negative Sign in Algebra

The negative sign in algebra changes the entire expression it is attached to. Placing a negative sign in front of a number or expression signifies the opposite value of that number.
In our given example: -\((5)^3\)Once you compute \(5^3 = 125\), the negative sign before this power indicates the opposite of 125, which is -125. Pay careful attention to the placement of negative signs, especially with powers, as they can significantly alter the outcome.

• When you see \(-x\), it means you get the negative of whatever \(x\) is.
• For example, if \(x = 5\), then \(-x = -5\).

It's essential to acknowledge that the negative sign's position is crucial.
For instance:

• \((-x)^3\) means the cube of \(-x\).
• Whereas, \(-x^3\) means the negative of \(x^3\).

Understanding how to correctly interpret and apply negative signs makes a big difference in solving algebraic problems and ensures accuracy in your calculations.