Problem 5 - Algebra 2

Question

Solve each equation. $$ \frac{1}{6}(12 a)^{\frac{1}{3}}=1 $$

Step-by-Step Solution

Answer

The solution is \(a = 18\).

Step 1: Isolate the Expression

The given equation is \( \frac{1}{6}(12a)^{\frac{1}{3}}=1 \). First, isolate the expression \((12a)^{\frac{1}{3}}\) by multiplying both sides of the equation by 6. This gives us \((12a)^{\frac{1}{3}} = 6\).

Step 2: Remove the Cube Root

To remove the cube root, raise both sides of the equation to the power of 3. This step yields \(12a = 6^3\). Calculate \(6^3\) to get \(216\), resulting in \(12a=216\).

Step 3: Solve for \(a\)

Divide both sides of the equation \(12a = 216\) by 12 to solve for \(a\). This results in \(a=\frac{216}{12}=18\).

Key Concepts

Isolation of Terms

When solving equations, a common first step is isolating the term containing the variable. This helps us focus on the main expression that we need to deal with to find the solution. In this equation, the term to isolate is \((12a)^{\frac{1}{3}}\), which is a cube root expression multiplied by \(\frac{1}{6}\). To eliminate the fraction and isolate \((12a)^{\frac{1}{3}}\), we multiply both sides of the equation by 6. This way we remove the fraction and focus on the cube root. Things to remember:

• Determine what expression needs isolating.
• Use inverse operations, like multiplication or division, to isolate it.
• Be careful to perform the operation on both sides of the equation to maintain balance.

Approaching equations this way sets up a clear and manageable path to solve for the variable. Remember, isolating terms can make challenging problems easier to tackle.

Cube Root

A cube root operation is the inverse of exponentiation to the power of 3. In our equation, once we have isolated \((12a)^{\frac{1}{3}}\), we need to remove the cube root to make further progress. The cube root is represented by the exponent \(\frac{1}{3}\). To cancel out the cube root, we raise the expression to the power of 3. Thinking about cube roots:

• Cube roots undo cubing (raising to the third power).
• Neutralizing a cube root requires raising the expression to the power of 3.

This action converts our expression to a simple linear form, where \(12a = 6^3 = 216\). By removing the cube root, we transition from a more complex operation to a standard multiplication problem.

Exponentiation

Exponentiation is a powerful mathematical operation, involving raising numbers to specific powers. It appears often in algebra to express repeated multiplication. In our problem, after isolating the term and raising it to the power of 3, we handle the number 6 raised to the power of 3 — written as \(6^3\). Points about exponentiation:

• \(6^3\) represents \(6 \times 6 \times 6\).
• Exponentiation simplifies repeated multiplication into a concise expression.
• The operation is fundamental in reversing roots as multiplying a number by itself a certain number of times.

Calculating \(6^3\) results in \(216\), which simplifies our equation to \(12a = 216\). Exponentiation thus transforms cube roots, making equations linear and easier to resolve.

Division in Equations

When equations simplify to a basic linear form, division is usually the final step to solve for the variable. In our example, we arrived at \(12a = 216\). Here, division comes into play to isolate the variable \(a\). Dividing both sides by 12 results in solving the equation. Key considerations for division in equations:

• Identify what number the variable is being multiplied by — here it's 12.
• Divide both sides by this number to bring the variable to a coefficient of 1.
• Ensure that division is performed on all terms to maintain equation balance.

This gives us \(a = \frac{216}{12} = 18\). Division is a straightforward, yet critical step that finalizes the isolation and solution for the variable.