Question

Solve the equation using inverse operations. Check your solutions. In your final answer, include all of your work.
2 + x^3 = 18


WRITER

Answer 1

To solve the equation \( 2 + x^3 = 18 \) using inverse operations, we will isolate \( x^3 \) first.

### Step 1: Isolate the term with \( x^3 \)

We start with the equation:
\[ 2 + x^3 = 18 \]

To isolate \( x^3 \), we subtract 2 from both sides:
\[
x^3 = 18 - 2
\]
\[
x^3 = 16
\]

### Step 2: Apply the inverse operation to solve for \( x \)

Now we need to find \( x \). Since \( x^3 \) means \( x \) raised to the power of 3, we will take the cube root of both sides:
\[
x = \sqrt[3]{16}
\]

### Step 3: Simplify the cube root

The value of \( \sqrt[3]{16} \) can be expressed as:
\[
x = \sqrt[3]{2^4}
\]
Using the property of exponents:
\[
x = 2^{4/3}
\]
This can also be written as:
\[
x = 2 \cdot 2^{1/3} = 2 \sqrt[3]{2}
\]

### Step 4: Check the solution

To verify our solution, we will substitute \( x = 2 \sqrt[3]{2} \) back into the original equation to ensure both sides are equal.

Start with the original equation:
\[
2 + x^3 = 18
\]

Substituting the value of \( x \):
\[
2 + (2 \sqrt[3]{2})^3 = 18
\]

Calculating \( (2 \sqrt[3]{2})^3 \):
\[
(2 \sqrt[3]{2})^3 = 2^3 \cdot (\sqrt[3]{2})^3 = 8 \cdot 2 = 16
\]

Now substituting back:
\[
2 + 16 = 18
\]
\[
18 = 18
\]

Since both sides of the equation are equal, our solution is confirmed.

### Final Answer

Thus, the solution to the equation \( 2 + x^3 = 18 \) is:
\[
x = \sqrt[3]{16} \quad \text{or} \quad x = 2 \cdot 2^{1/3}
\]