Problem 35 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ \sqrt[3]{7 n-1}=3 $$

Step-by-Step Solution

Answer

The solution is \( n = 4 \).

Step 1: Understand the problem statement

We need to solve for the variable \( n \) in the equation \( \sqrt[3]{7n-1}=3 \). This means we must find the value of \( n \) that makes this equation true.

Step 2: Isolate the cube root

The given equation is \( \sqrt[3]{7n-1}=3 \). In this step, let's keep the expression inside the cube root isolated on one side by leaving the equation as is since it is already isolated.

Step 3: Eliminate the cube root

To remove the cube root, we need to cube both sides of the equation. Cubing both sides gives us: \( (\sqrt[3]{7n-1})^3 = 3^3 \). This simplifies to \( 7n - 1 = 27 \).

Step 4: Solve for n

Now, solve for \( n \) in the equation \( 7n - 1 = 27 \). First, add 1 to both sides to get \( 7n = 28 \). Next, divide both sides by 7 to isolate \( n \): \( n = \frac{28}{7} \).

Step 5: Simplify the solution

Finally, simplify \( \frac{28}{7} \) to get \( n = 4 \).

Key Concepts

Cube root

The cube root is a mathematical operation where we find a number that, when multiplied by itself twice more (a total of three factors), gives the original number. In our equation, we see a cube root symbol: \( \sqrt[3]{7n-1} \). This tells us that whatever is inside the cube root is the number we want to multiply by itself three times to reach the number 3. Cube roots are essential in solving equations involving cubic relationships.

• For the cube root \( \sqrt[3]{x} \), \( x \) is called the radicand.
• The result of \( \sqrt[3]{x} \) is a number which when cubed gives \( x \).

Understanding cube roots helps in reversing cubing operations, as cubing and taking the cube root are inverse operations.

Isolating variables

Isolating a variable in an equation means rearranging the equation so that the variable is on one side and everything else is on the opposite side. In the given equation \( \sqrt[3]{7n-1} = 3 \), the expression inside the cube root \( 7n-1 \) is already isolated. This step involves setting up the equation so that the variable changes can be straightforwardly made.

• Rearrange terms to "uncover" variable terms.
• Combine like terms or use inverse operations.

Isolating the variable prepares it for further operations like removing cube roots and solving the equation.

Cubing

Cubing is the process of raising a number to the power of three. In other words, it's multiplying a number by itself and by itself again. In the solve process, after isolating, the expression in the cube root, we cubed both sides to remove the cube root. We had \( (\sqrt[3]{7n-1})^3 = 3^3 \). This process allowed us to eliminate the cube root and simplify the equation further.

• When you cube a cube root, you simply get the expression inside back.
• Cubing both sides is used to maintain equality while eliminating the cube root.

Cubing is a powerful tool when working with cube roots, as it helps break down into simpler steps.

Simplifying equations

Simplifying equations is the process of making them easier to solve or understand. After eliminating the cube root by cubing both sides, we continued by simplifying \( 7n - 1 = 27 \) into \( 7n = 28 \), and finally solving for \( n \). Simplification is key to reducing complex equations into more manageable forms.

• Combine like terms: This means adding or subtracting similar terms on both sides.
• Perform operations such as adding, subtracting, multiplying, or dividing to isolate variables.

Each step in simplification brings us closer to the solution, making sure the equation's integrity remains intact.