Problem 16 - A Graphical Approach to College Algebra

Question

Solve each equation by hand. Do not use a calculator. $$\sqrt[3]{x+9}=2$$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Understand the Cube Root

The given equation is \(\sqrt[3]{x+9} = 2\). This equation implies that \(x+9\) is the number whose cube root is equal to 2.

Step 2: Eliminate the Cube Root

To eliminate the cube root, we need to cube both sides of the equation. If we cube both sides, we get \((\sqrt[3]{x+9})^3 = 2^3\), which simplifies to \(x+9 = 8\), since cubing the cube root removes the root.

Step 3: Solve for x

Now, solve the equation \(x + 9 = 8\) for \(x\). Subtract 9 from both sides to isolate \(x\): \[ x = 8 - 9 \] \[ x = -1 \].

Key Concepts

Cube Root

The cube root of a number is similar to the concept of a square root. While a square root refers to a number that, when multiplied by itself, gives the original number, a cube root refers to a number that, when used in a multiplication three times, results in the original number. The cube root of a number \( a \) is denoted as \( \sqrt[3]{a} \).
For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \). Similarly, the problem given \( \sqrt[3]{x+9} = 2 \) indicates that the expression inside the root, \( x+9 \), raised to the power of three equals 2 squared three times.
This is an important operation in algebra, as it allows us to simplify equations involving cube roots by converting them back into easy-to-handle expressions.

Step-by-Step Solution

Breaking down complex problems into smaller steps makes solving them much easier. The problem \( \sqrt[3]{x+9} = 2 \) can be approached methodically to understand each move. Follow these steps:
1. **Understand the Equation**: Recognize that the expression represents the cube root and identify the value that needs to be isolated within the root.
2. **Eliminate the Cube Root**: To remove the cube root, cube both sides of the equation. This gives \( (\sqrt[3]{x+9})^3 = 2^3 \), simplifying to \( x+9 = 8 \).
3. **Solve for \( x \)**: It’s easier now to subtract 9 from both sides, leading to \( x = 8 - 9 \). This results in \( x = -1 \).
Each step supports the previous one, ensuring the solution unfolds logically and clearly.

Isolating Variables

Isolating the variable is an essential skill in algebra, enabling us to find its value. When facing the equation \( x+9 = 8 \), the goal is to solve for \( x \).
To isolate \( x \), you need to remove the 9 added to it. Achieve this by performing the same operation on both sides of the equation: subtract 9. So, \( x+9 - 9 \) becomes \( x \) alone on one side, while 8 turns into 8 - 9 on the other, simplifying to \( x = -1 \).
This technique—modifying both sides of an equation equally—keeps the equation balanced. It's like a scale, needing equal weight on either side to maintain equilibrium. This step completes the process of solving for the variable, rounding off your efforts to determine its value.