Problem 43 - Algebra 1

Question

Evaluate the expression. $$y^{3}-4 \text { when } y=-2$$

Step-by-Step Solution

Answer

The evaluated expression is -12.

Step 1: Substitute y with -2

Now, substitute -2 for \(y\) in the expression \(y^{3}-4\). This gives us \((-2)^{3}-4\).

Step 2: Simplify the equation

The next step is to simplify the equation. Notice that \((-2)^{3} = -8\). Therefore the expression becomes \(-8-4\).

Step 3: Calculate Final Answer

Finally, if we continue the simplification, the equation results to -12.

Key Concepts

Exponents

In mathematics, exponents are used to indicate how many times a number should be multiplied by itself. When we have something like \(y^3\), it's a way of saying you should multiply \(y\) by itself three times. For instance, if \(y = -2\), calculating \((-2)^3\) means \((-2) \times (-2) \times (-2)\). Do this step-by-step:

• First, multiply \((-2) \times (-2)\) to get \(4\).
• Next, multiply that result by \(-2\) again, resulting in \(-8\).

Remember to focus on both the number and its sign, especially with negative bases. Exponents can change how the multiplication affects the signs.

Negative Numbers

Negative numbers can sometimes seem tricky, but understanding how they work is crucial. When dealing with negative bases raised to an exponent, like \((-2)^3\), remember:

• Exponents indicate the number of times the base multiplies by itself.
• When a negative number is raised to an odd exponent, the result is negative. For example, \((-2)^3\) yields \(-8\).
• Conversely, positive results occur with even exponents, e.g., \((-2)^2 = 4\).

Using negative values requires careful attention to signs, especially during multiplication, to avoid common pitfalls.

Substitution

Substitution is a simple yet powerful tool in algebra. It involves replacing a variable with a given number value. For the expression \(y^3 - 4\), and knowing \(y = -2\), you substitute \(-2\) in place of \(y\), making your expression \((-2)^3 - 4\). This often simplifies complex expressions, allowing you to focus on calculating the result step-by-step. The process helps in solving equations by converting an expression with variables into an arithmetic problem.

Simplification

Simplification involves reducing an expression to its simplest form, making calculations easier and results clear. With the problem \((-2)^3 - 4\), we simplified by first evaluating the exponent:

• Calculate \((-2)^3\) to get \(-8\).
• Then, subtract \(4\), leading to \(-8 - 4 = -12\).

By following basic arithmetic rules, you break down each step into manageable parts, ensuring accuracy and clarity. Simplification is crucial in solving mathematical problems efficiently and helps in understanding the core operations within an expression.