If `r^(3)=1728` then find r

Options

A. 12
B. 14
C. 16
D. 18

Answer

To solve the equation \( r^3 = 1728 \), we need to find the value of \( r \). Here’s a step-by-step solution:

### Step 1: Write the equation
We start with the equation:
\[ r^3 = 1728 \]

### Step 2: Take the cube root
To find \( r \), we need to take the cube root of both sides of the equation:
\[ r = \sqrt[3]{1728} \]

### Step 3: Factor 1728
Next, we will factor 1728 to find its prime factors. We can start dividing by 2:
- \( 1728 \div 2 = 864 \)
- \( 864 \div 2 = 432 \)
- \( 432 \div 2 = 216 \)
- \( 216 \div 2 = 108 \)
- \( 108 \div 2 = 54 \)
- \( 54 \div 2 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, the prime factorization of 1728 is:
\[ 1728 = 2^6 \times 3^3 \]

### Step 4: Rewrite the cube root
Now we can rewrite the cube root using the prime factorization:
\[ r = \sqrt[3]{2^6 \times 3^3} \]

### Step 5: Simplify the cube root
Using the property of exponents, we can simplify:
\[ r = \sqrt[3]{2^6} \times \sqrt[3]{3^3} = 2^{6/3} \times 3^{3/3} = 2^2 \times 3^1 \]
\[ r = 4 \times 3 = 12 \]

### Conclusion
Thus, the value of \( r \) is:
\[ \boxed{12} \]

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