The simplest rationalisation factor of `""^(3)sqrt(500)` is

Options

A. `sqrt(5)`
B. `""^(3)sqrt(2)`
C. `""^(3)sqrt(5)`
D. `1/(""^(3)sqrt(2))`

Answer

To find the simplest rationalization factor of \( \sqrt[3]{500} \), we can follow these steps:

### Step 1: Factor 500
We start by factoring 500 into its prime factors:
\[ 500 = 5^3 \times 2^2 \]

### Step 2: Rewrite the Cube Root
Now we can express the cube root of 500 using its prime factors:
\[ \sqrt[3]{500} = \sqrt[3]{5^3 \times 2^2} \]

### Step 3: Apply the Property of Cube Roots
Using the property of cube roots, we can separate the factors:
\[ \sqrt[3]{500} = \sqrt[3]{5^3} \times \sqrt[3]{2^2} \]

### Step 4: Simplify the Cube Root
Now we simplify the expression:
\[ \sqrt[3]{5^3} = 5 \]
Thus, we have:
\[ \sqrt[3]{500} = 5 \times \sqrt[3]{2^2} = 5 \times \sqrt[3]{4} \]

### Step 5: Rationalization Factor
To rationalize \( \sqrt[3]{4} \), we need to multiply it by \( \sqrt[3]{2} \) so that we can express it as a whole number:
\[ \text{Rationalization factor} = \sqrt[3]{2} \]

### Final Answer
Thus, the simplest rationalization factor of \( \sqrt[3]{500} \) is:
\[ \sqrt[3]{2} \]

---