Problem 4 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ 4 w^{3}+7=0 $$

Step-by-Step Solution

Answer

Answer: The value of w is w = ∛(-7/4).

Step 1: Subtract 7 from both sides of the equation

We need to simplify the equation before solving it. We can do this by subtracting 7 from both sides of the equation: $$ 4w^3 + 7 - 7 = 0 - 7 $$

Step 2: Simplify the equation

Now, we can simplify the equation by doing the subtraction from the previous step: $$ 4w^3 = -7 $$

Step 3: Divide both sides by 4

To solve for w, we need to isolate w. We can do this by dividing both sides of the equation by 4: $$ \frac{4w^3}{4} = \frac{-7}{4} $$

Step 4: Simplify the equation

Now, we can simplify the equation by performing the division from the previous step: $$ w^3 = -\frac{7}{4} $$

Step 5: Take the cube root of both sides

Now, we need to find the value of w by taking the cube root of both sides of the equation: $$ \sqrt[3]{w^3} = \sqrt[3]{-\frac{7}{4}} $$

Step 6: Simplify and find the solution

Finally, simplify the equation and find the value of w that satisfies the equation: $$ w = \sqrt[3]{-\frac{7}{4}} $$ The solution to the equation is: $$ w = \sqrt[3]{-\frac{7}{4}} $$

Key Concepts

Isolating Variables

When solving an equation, one of the primary goals is to isolate the variable you are solving for. This means you want the variable to be by itself on one side of the equation. In our given exercise, we're solving for the variable \( w \) in the equation \( 4w^3 + 7 = 0 \). To begin, we need to eliminate any numbers or coefficients that are with the variable. This process involves basic arithmetic operations like addition or subtraction.

• Start by moving constants to the other side of the equation. For example, subtracting \( 7 \) from both sides removes the constant on the left side, simplifying the equation to \( 4w^3 = -7 \).
• Next, eliminate coefficients attached to the variable by performing inverse operations. Here, divide by \( 4 \) to get \( w^3 = -\frac{7}{4} \). This process allows you to successfully isolate \( w^3 \).

Isolating the variable is crucial as it reduces the equation to a simpler form, making it easier to solve the next steps.

Cube Root

The cube root is a mathematical function that reverses the operation of cubing a number. If you have a variable raised to the power of three, like \( w^3 \), taking the cube root will help you solve for the variable itself. In our problem, once we have isolated \( w^3 \), we take the cube root to find \( w \).

• The cube root is represented by a small 3 written above the root symbol: \( \sqrt[3]{\cdot}\). Taking the cube root of \( w^3 \) will give us \( w \).
• Apply this principle to both sides of your equation: \( \sqrt[3]{w^3} = \sqrt[3]{-\frac{7}{4}} \). This simplifies to \( w = \sqrt[3]{-\frac{7}{4}} \).

Taking the cube root helps in directly finding the variable, bringing you closer to the solution.

Equation Simplification

Simplifying an equation means breaking it down to its most basic form where no further operations are needed. Every step we performed thus far, like moving terms across the equal sign and dividing, is part of equation simplification. Through simplification, we ease the pathway toward finding solutions.

• Begin by ensuring your equation is clean of additional terms, which involves moving constants and simplifying expressions as seen in \( 4w^3 = -7 \).
• Continuing with \( w^3 = -\frac{7}{4} \), we've divided by \( 4 \), effectively reducing the equation to its simplest form before finally solving for \( w \).

This simplification makes it much easier to take further operations like cube roots, leading you straightforwardly to the solution \( w = \sqrt[3]{-\frac{7}{4}} \). A simplified equation is cleaner and more straightforward to interpret, eliminating chances of errors.