`root3(0.125)+root3(0.729)=n/10.` Find n.

Answer

To solve the equation \( \sqrt[3]{0.125} + \sqrt[3]{0.729} = \frac{n}{10} \), we will follow these steps:

### Step 1: Rewrite the decimal numbers as fractions
First, we convert the decimal numbers into fractions:
- \( 0.125 = \frac{125}{1000} \)
- \( 0.729 = \frac{729}{1000} \)

### Step 2: Apply the cube root to the fractions
Now we can express the cube roots:
- \( \sqrt[3]{0.125} = \sqrt[3]{\frac{125}{1000}} \)
- \( \sqrt[3]{0.729} = \sqrt[3]{\frac{729}{1000}} \)

### Step 3: Simplify the cube roots
Using the property of cube roots, we can separate the numerator and denominator:
- \( \sqrt[3]{\frac{125}{1000}} = \frac{\sqrt[3]{125}}{\sqrt[3]{1000}} \)
- \( \sqrt[3]{\frac{729}{1000}} = \frac{\sqrt[3]{729}}{\sqrt[3]{1000}} \)

Since \( \sqrt[3]{1000} = 10 \):
- \( \sqrt[3]{0.125} = \frac{\sqrt[3]{125}}{10} \)
- \( \sqrt[3]{0.729} = \frac{\sqrt[3]{729}}{10} \)

### Step 4: Calculate the cube roots of 125 and 729
Now we find the cube roots:
- \( \sqrt[3]{125} = 5 \) (since \( 5^3 = 125 \))
- \( \sqrt[3]{729} = 9 \) (since \( 9^3 = 729 \))

### Step 5: Substitute back into the equation
Substituting these values back, we have:
\[ \sqrt[3]{0.125} + \sqrt[3]{0.729} = \frac{5}{10} + \frac{9}{10} \]

### Step 6: Combine the fractions
Now, we can combine the fractions:
\[ \frac{5}{10} + \frac{9}{10} = \frac{5 + 9}{10} = \frac{14}{10} \]

### Step 7: Set the equation equal to \(\frac{n}{10}\)
Now we have:
\[ \frac{14}{10} = \frac{n}{10} \]

### Step 8: Solve for \(n\)
Since the denominators are the same, we can equate the numerators:
\[ n = 14 \]

Thus, the value of \(n\) is \(14\).

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