Problem 108 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ y^{3}+3 x $$

Step-by-Step Solution

Answer

The evaluated expression is -42.

Step 1: Substitute the Variables

Begin by substituting the given values into the expression. Replace \(x\) with \(-5\) and \(y\) with \(-3\). The expression becomes \[(-3)^3 + 3(-5)\]

Step 2: Calculate the Exponents

Evaluate the exponent portion of the expression. Here, we calculate \((-3)^3\):\[(-3)^3 = -27\]Thus, the expression becomes\[-27 + 3(-5)\]

Step 3: Compute the Multiplication

Next, compute the multiplication part of the expression: \[3(-5) = -15\]Now, substitute back into the expression:\[-27 + (-15)\]

Step 4: Sum the Numbers

Add \(-27\) and \(-15\) together:\[-27 + (-15) = -42\]Thus, the evaluated expression equals \(-42\).

Key Concepts

Substituting Values

In algebra, one of the most common tasks is evaluating expressions using given variable values. This is called "substituting values". It's akin to replacing placeholders in a formula with actual numbers. Let's consider the expression \( y^3 + 3x \) and the given values \( x=-5 \) and \( y=-3 \).

• Start by substituting \(-5\) for \(x\) and \(-3\) for \(y\).
• This turns the expression into \((-3)^3 + 3(-5)\).

Substituting values helps us simplify the expression, making it easier to compute and understand. By following this process, each part of the expression that involved a variable is now a specific number, ready for further operations.

Exponentiation

Exponentiation is the process of raising a number to a power. It's a shortcut for multiplication. When we have \( (-3)^3 \), it means multiplying \(-3\) by itself three times.

• So, \((-3) \times (-3) = 9\).
• Multiply again: \(9 \times (-3) = -27\).

Thus, \((-3)^3 = -27\). Notice that an odd exponent of a negative number results in a negative product. Exponentiation is essential in algebra for simplifying expressions and finding exact values quickly.

Order of Operations

When evaluating expressions, the order of operations is crucial. It tells us which computations to perform first. The rule can be remembered using "PEMDAS":

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

In our expression \((-3)^3 + 3(-5)\), we address the exponent first: \((-3)^3 = -27\).
Next, we perform the multiplication: \(3(-5) = -15\).
Finally, we add these results: \(-27 + (-15) = -42\). Following the order of operations ensures we solve problems accurately without errors.

Integer Arithmetic

Integer arithmetic refers to operations involving whole numbers, which can be positive, negative, or zero. In our example, we deal with the arithmetic of negative integers.

• Multiplication of a positive and a negative integer results in a negative product: \(3(-5) = -15\).
• Adding two negative integers combines their absolute values and keeps the negative sign: \(-27 + (-15) = -42\).

Understanding integer arithmetic is crucial, especially in evaluating expressions. It allows us to handle complexities like negative signs with confidence and precision.