`(root(3)(1000) + root(3)(0.008) - root(3)(0.125))` is equal to

Options

A. 9.7
B. 9.97
C. 9.997
D. 9.9997

Answer

To solve the expression \( \sqrt[3]{1000} + \sqrt[3]{0.008} - \sqrt[3]{0.125} \), we will evaluate each term step by step.

### Step 1: Calculate \( \sqrt[3]{1000} \)
The cube root of 1000 can be calculated as follows:
\[ \sqrt[3]{1000} = 10 \]
This is because \( 10^3 = 1000 \).

**Hint:** Remember that the cube root of a number \( x \) is the number \( y \) such that \( y^3 = x \).

### Step 2: Calculate \( \sqrt[3]{0.008} \)
Next, we calculate the cube root of 0.008:
\[ \sqrt[3]{0.008} = \sqrt[3]{\frac{8}{1000}} = \frac{\sqrt[3]{8}}{\sqrt[3]{1000}} = \frac{2}{10} = 0.2 \]
Here, \( \sqrt[3]{8} = 2 \) and \( \sqrt[3]{1000} = 10 \).

**Hint:** To find the cube root of a fraction, take the cube root of the numerator and the denominator separately.

### Step 3: Calculate \( \sqrt[3]{0.125} \)
Now, we find the cube root of 0.125:
\[ \sqrt[3]{0.125} = \sqrt[3]{\frac{125}{1000}} = \frac{\sqrt[3]{125}}{\sqrt[3]{1000}} = \frac{5}{10} = 0.5 \]
Here, \( \sqrt[3]{125} = 5 \) and \( \sqrt[3]{1000} = 10 \).

**Hint:** Similar to the previous step, break down the fraction to find the cube roots of the numerator and denominator.

### Step 4: Combine the results
Now we can substitute back into the original expression:
\[ \sqrt[3]{1000} + \sqrt[3]{0.008} - \sqrt[3]{0.125} = 10 + 0.2 - 0.5 \]

### Step 5: Perform the arithmetic
Now we perform the arithmetic:
\[ 10 + 0.2 = 10.2 \]
Then,
\[ 10.2 - 0.5 = 9.7 \]

### Final Answer
Thus, the value of the expression \( \sqrt[3]{1000} + \sqrt[3]{0.008} - \sqrt[3]{0.125} \) is:
\[ \boxed{9.7} \]