Difference of two perfect cubes is 189. If the cube root of the smaller of the two numbers is 3, find the cube root of the larger number.

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem step-by-step, we will follow these instructions:

### Step 1: Understand the given information
We know that the difference of two perfect cubes is 189, and the cube root of the smaller number is 3.

### Step 2: Define the variables
Let:
- \( B \) be the smaller perfect cube.
- \( A \) be the larger perfect cube.

### Step 3: Calculate the value of \( B \)
Since the cube root of the smaller number \( B \) is 3, we can find \( B \) by cubing 3:
\[ B = 3^3 = 27 \]

### Step 4: Set up the equation for the difference of cubes
According to the problem, the difference between the two cubes is:
\[ A - B = 189 \]
Substituting the value of \( B \) we found:
\[ A - 27 = 189 \]

### Step 5: Solve for \( A \)
Now, we can solve for \( A \):
\[ A = 189 + 27 = 216 \]

### Step 6: Find the cube root of \( A \)
Now we need to find the cube root of the larger number \( A \):
\[ \text{Cube root of } A = A^{1/3} = 216^{1/3} \]
To find \( 216^{1/3} \), we can factor 216:
\[ 216 = 6 \times 6 \times 6 = 6^3 \]
Thus,
\[ 216^{1/3} = 6 \]

### Final Answer
The cube root of the larger number is \( 6 \).

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