Find the value of `root(3)(27)xxroot(3)(216)xxroot(3)(64)`.

Options

A. 24
B. 45
C. 72
D. 96

Answer

To find the value of \( \sqrt[3]{27} \times \sqrt[3]{216} \times \sqrt[3]{64} \), we can follow these steps:

### Step 1: Break down the numbers into their prime factors
- **27** can be factored as:
\[ 27 = 3 \times 3 \times 3 = 3^3 \]
- **216** can be factored as:
\[ 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3 \]
- **64** can be factored as:
\[ 64 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = (2^3)^2 = 4^3 \]

### Step 2: Rewrite the expression using the prime factors
Now we can rewrite the expression:
\[ \sqrt[3]{27} \times \sqrt[3]{216} \times \sqrt[3]{64} = \sqrt[3]{3^3} \times \sqrt[3]{2^3 \times 3^3} \times \sqrt[3]{4^3} \]

### Step 3: Apply the property of cube roots
Using the property \( \sqrt[3]{a^3} = a \), we can simplify:
\[ \sqrt[3]{3^3} = 3 \]
\[ \sqrt[3]{2^3 \times 3^3} = \sqrt[3]{2^3} \times \sqrt[3]{3^3} = 2 \times 3 = 6 \]
\[ \sqrt[3]{4^3} = 4 \]

### Step 4: Multiply the results together
Now we can multiply the results:
\[ 3 \times 6 \times 4 \]

### Step 5: Calculate the final result
Calculating step-by-step:
- First, calculate \( 3 \times 6 = 18 \)
- Then, calculate \( 18 \times 4 = 72 \)

Thus, the final value is:
\[ \sqrt[3]{27} \times \sqrt[3]{216} \times \sqrt[3]{64} = 72 \]

### Final Answer:
The value of \( \sqrt[3]{27} \times \sqrt[3]{216} \times \sqrt[3]{64} \) is **72**.

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