`root3(3.43)/root(3)(10)=………….`

Answer

To solve the expression \( \frac{\sqrt[3]{3.43}}{\sqrt[3]{10}} \), we will follow these steps:

### Step 1: Convert the decimal to a fraction
We can express \( 3.43 \) as \( \frac{343}{100} \). This is because moving the decimal two places to the right gives us \( 343 \), and we divide by \( 100 \) to maintain the value.

**Hint:** To convert a decimal to a fraction, count how many places the decimal moves to the right and use that as the denominator (e.g., \( 0.01 \) becomes \( \frac{1}{100} \)).

### Step 2: Rewrite the expression
Now we can rewrite the expression as:
\[ \frac{\sqrt[3]{\frac{343}{100}}}{\sqrt[3]{10}} \]

**Hint:** When dividing cube roots, you can combine them into one cube root: \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}} \).

### Step 3: Combine the cube roots
This gives us:
\[ \sqrt[3]{\frac{343}{100 \cdot 10}} = \sqrt[3]{\frac{343}{1000}} \]

**Hint:** When multiplying fractions, multiply the numerators together and the denominators together.

### Step 4: Simplify the cube root
Next, we recognize that \( 1000 = 10^3 \), so we can rewrite the expression as:
\[ \sqrt[3]{\frac{343}{10^3}} = \frac{\sqrt[3]{343}}{\sqrt[3]{1000}} \]

**Hint:** Remember that \( \sqrt[3]{a^3} = a \).

### Step 5: Calculate the cube roots
We know that \( 343 = 7^3 \), so \( \sqrt[3]{343} = 7 \), and \( \sqrt[3]{1000} = 10 \).

Thus, we have:
\[ \frac{7}{10} \]

**Hint:** To find the cube root of a number, look for perfect cubes or factor the number into prime factors.

### Final Answer
The final result is:
\[ \frac{7}{10} \]