`(root3(1.728)-root3(0.216))/(root3(2.197)-root3(0.343))=`

Options

A. 1
B. 1
C. 2
D. -2

Answer

To solve the expression \((\sqrt[3]{1.728} - \sqrt[3]{0.216}) / (\sqrt[3]{2.197} - \sqrt[3]{0.343})\), we can follow these steps:

### Step 1: Rewrite the numbers in fractional form
We can express the numbers under the cube roots as fractions:
- \(1.728 = \frac{1728}{1000}\)
- \(0.216 = \frac{216}{1000}\)
- \(2.197 = \frac{2197}{1000}\)
- \(0.343 = \frac{343}{1000}\)

Thus, we can rewrite the expression as:
\[ \frac{\sqrt[3]{\frac{1728}{1000}} - \sqrt[3]{\frac{216}{1000}}}{\sqrt[3]{\frac{2197}{1000}} - \sqrt[3]{\frac{343}{1000}}} \]

### Step 2: Factor out the common cube root
We can factor out \(\sqrt[3]{1000}\) from both the numerator and the denominator:
\[ = \frac{\frac{1}{\sqrt[3]{1000}} (\sqrt[3]{1728} - \sqrt[3]{216})}{\frac{1}{\sqrt[3]{1000}} (\sqrt[3]{2197} - \sqrt[3]{343})} \]

This simplifies to:
\[ = \frac{\sqrt[3]{1728} - \sqrt[3]{216}}{\sqrt[3]{2197} - \sqrt[3]{343}} \]

### Step 3: Calculate the cube roots
Now we can calculate the cube roots:
- \(\sqrt[3]{1728} = 12\) (since \(12^3 = 1728\))
- \(\sqrt[3]{216} = 6\) (since \(6^3 = 216\))
- \(\sqrt[3]{2197} = 13\) (since \(13^3 = 2197\))
- \(\sqrt[3]{343} = 7\) (since \(7^3 = 343\))

### Step 4: Substitute the values back into the expression
Now substituting these values back, we have:
\[ = \frac{12 - 6}{13 - 7} \]

### Step 5: Simplify the expression
Now, simplify the numerator and denominator:
- Numerator: \(12 - 6 = 6\)
- Denominator: \(13 - 7 = 6\)

Thus, we have:
\[ = \frac{6}{6} = 1 \]

### Final Answer
The value of the expression is \(1\).

---