Question

Solve the equation using inverse operations. Check your solutions. In your final answer, include all of your work. 1/4x^3 = -27/4






WRITER

Answer 1

To solve the equation \( \frac{1}{4} x^3 = -\frac{27}{4} \), we can use inverse operations. Here are the steps:

### Step 1: Eliminate the Fraction
To eliminate the \( \frac{1}{4} \) coefficient, we can multiply both sides of the equation by 4:

\[
4 \cdot \left( \frac{1}{4} x^3 \right) = 4 \cdot \left( -\frac{27}{4} \right)
\]

This simplifies to:

\[
x^3 = -27
\]

### Step 2: Take the Cube Root
Next, we want to isolate \( x \) by taking the cube root of both sides:

\[
x = \sqrt[3]{-27}
\]

Calculating the cube root of \(-27\):

\[
x = -3
\]

### Step 3: Check the Solution
To verify our solution, we will substitute \( x = -3 \) back into the original equation:

\[
\frac{1}{4} (-3)^3 = -\frac{27}{4}
\]

Calculating \((-3)^3\):

\[
(-3)^3 = -27
\]

Now substituting back:

\[
\frac{1}{4} \cdot (-27) = -\frac{27}{4}
\]

Thus, we find:

\[
-\frac{27}{4} = -\frac{27}{4}
\]

The left-hand side equals the right-hand side, confirming that our solution is correct.

### Final Answer
The solution to the equation \( \frac{1}{4} x^3 = -\frac{27}{4} \) is:

\[
\boxed{-3}
\]