Problem 7 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(3 \sqrt{n}-2=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(n = \frac{4}{9}\).

Step 1: Isolate the Radical Term

Start with the given equation: \(3 \sqrt{n} - 2 = 0\). Add 2 to both sides to isolate the term with the square root: \(3\sqrt{n} = 2\).

Step 2: Solve for the Square Root of n

Divide both sides by 3 to solve for \(\sqrt{n}\): \(\sqrt{n} = \frac{2}{3}\).

Step 3: Remove the Square Root

Square both sides to remove the square root and solve for \(n\): \((\sqrt{n})^2 = \left(\frac{2}{3}\right)^2\), which gives \(n = \frac{4}{9}\).

Step 4: Check the Solution

Substitute \(n = \frac{4}{9}\) back into the original equation to ensure it satisfies the equation. Calculate \(3\sqrt{\frac{4}{9}} - 2\) which simplifies to \(3 \times \frac{2}{3} - 2 = 2 - 2 = 0\). The left side equals the right side, confirming the solution is correct.

Key Concepts

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In the equation \(3 \sqrt{n} - 2 = 0\), the square root symbol \(\sqrt{}\) indicates we are looking for a number that, when squared, equals \(n\). Understanding this concept is vital for solving equations involving square roots.

To simplify expressions involving square roots:

• The square root of a perfect square (like 4 or 9) results in a whole number, such as \(\sqrt{4} = 2\).
• When dealing with fractions under a square root, remember \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\).

In the exercise, isolating \(\sqrt{n}\) was one of the first steps. This isolation helps us focus directly on solving for \(n\) itself.

Solving Equations

Solving an equation involves finding the value of the unknown variable that makes the equation true. In our example, we aim to find the value of \(n\) in \(3 \sqrt{n} - 2 = 0\). Understanding each operation's role is crucial for the correct solution.

Consider these steps in solving equations:

• Isolate the variable: In this context, we want to isolate \(\sqrt{n}\) by making one side of the equation solely dependent on it.
• Reverse operations: Undo any subtraction or addition first, then handle multiplication or division. Lastly, if necessary, square to remove the square root.

By applying these principles, we start with \(3 \sqrt{n} = 2\) and eventually solve for \(n\) by rearranging and squaring, ensuring every step precisely follows mathematical rules.

Step by Step Solution

Following a detailed step-by-step solution helps ensure accuracy and improve comprehension. With the original problem, starting with \(3 \sqrt{n} - 2 = 0\), students should follow these steps carefully:

1. **Isolate the radical term:** Begin by moving all non-radical components to the opposite side. Here, you add 2 to both sides to get \(3\sqrt{n} = 2\).2. **Solve for the square root:** Divide both sides by 3, simplifying to \(\sqrt{n} = \frac{2}{3}\).3. **Remove the square root:** To eliminate the square root, square both sides, yielding \(n = \frac{4}{9}\).4. **Verify the solution:** Substitute \(n = \frac{4}{9}\) back into the original equation to check if it holds true. By breaking it down as shown, learners can better understand each action's purpose and result.