The value of `3sqrt(500) xx 3sqrt(16)` is

Options

A. 16
B. 20
C. 25
D. 18

Answer

To solve the problem \(3\sqrt[3]{500} \times 3\sqrt[3]{16}\), we will follow these steps:

### Step 1: Combine the cube roots
Since both terms are cube roots, we can combine them under a single cube root:
\[ 3\sqrt[3]{500} \times 3\sqrt[3]{16} = 3 \times 3 \times \sqrt[3]{500 \times 16} \]

### Step 2: Calculate the multiplication inside the cube root
Now, we need to multiply \(500\) and \(16\):
\[ 500 \times 16 = 8000 \]

### Step 3: Rewrite the expression
Now, we can rewrite our expression:
\[ 3 \times 3 \times \sqrt[3]{8000} = 9 \times \sqrt[3]{8000} \]

### Step 4: Simplify the cube root
Next, we simplify \(\sqrt[3]{8000}\). We can express \(8000\) as:
\[ 8000 = 8 \times 1000 = 8 \times 10^3 \]
So, we can write:
\[ \sqrt[3]{8000} = \sqrt[3]{8 \times 10^3} = \sqrt[3]{8} \times \sqrt[3]{10^3} \]

### Step 5: Calculate the cube roots
We know that:
\[ \sqrt[3]{8} = 2 \quad \text{and} \quad \sqrt[3]{10^3} = 10 \]
Thus:
\[ \sqrt[3]{8000} = 2 \times 10 = 20 \]

### Step 6: Final multiplication
Now, we can substitute back into our expression:
\[ 9 \times \sqrt[3]{8000} = 9 \times 20 = 180 \]

### Final Answer
The value of \(3\sqrt[3]{500} \times 3\sqrt[3]{16}\) is \(180\).

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