Calculate the sub-triplicate ratio of 512 : 729.

Options

A. `9 : 8`
B. `4 :9`
C. `7 :8`
D. `8 :9`

Answer

To calculate the sub-triplicate ratio of 512 : 729, we will follow these steps:

### Step 1: Understand the concept of sub-triplicate ratio
The sub-triplicate ratio of two numbers is found by taking the cube root of the ratio of those two numbers.

### Step 2: Write the ratio as a fraction
We start with the ratio 512 : 729, which can be written as:
\[ \frac{512}{729} \]

### Step 3: Take the cube root of the ratio
To find the sub-triplicate ratio, we need to take the cube root of the fraction:
\[ \sqrt[3]{\frac{512}{729}} = \frac{\sqrt[3]{512}}{\sqrt[3]{729}} \]

### Step 4: Calculate the cube roots of the numerator and denominator
- The cube root of 512 is:
\[ \sqrt[3]{512} = 8 \quad \text{(since } 8^3 = 512\text{)} \]
- The cube root of 729 is:
\[ \sqrt[3]{729} = 9 \quad \text{(since } 9^3 = 729\text{)} \]

### Step 5: Write the sub-triplicate ratio
Now we can substitute the cube roots back into the fraction:
\[ \frac{\sqrt[3]{512}}{\sqrt[3]{729}} = \frac{8}{9} \]

### Final Answer
Thus, the sub-triplicate ratio of 512 : 729 is:
\[ 8 : 9 \]

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