Calculate the value of `root3(192/81)`

Options

A. `5/3`
B. `4/3`
C. `3/2`
D. `13/9`

Answer

To calculate the value of \( \sqrt[3]{\frac{192}{81}} \), we can follow these steps:

### Step 1: Factorize the numbers
First, we need to factorize both the numerator (192) and the denominator (81).

- **Factorization of 192**:
- 192 is divisible by 2: \( 192 \div 2 = 96 \)
- 96 is divisible by 2: \( 96 \div 2 = 48 \)
- 48 is divisible by 2: \( 48 \div 2 = 24 \)
- 24 is divisible by 2: \( 24 \div 2 = 12 \)
- 12 is divisible by 2: \( 12 \div 2 = 6 \)
- 6 is divisible by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.

So, the prime factorization of 192 is:
\[ 192 = 2^6 \times 3^1 \]

- **Factorization of 81**:
- 81 is \( 3 \times 3 \times 3 \times 3 = 3^4 \).

### Step 2: Write the fraction with its prime factors
Now we can express the fraction \( \frac{192}{81} \) using its prime factors:
\[ \frac{192}{81} = \frac{2^6 \times 3^1}{3^4} \]

### Step 3: Simplify the fraction
We can simplify the fraction:
\[ \frac{192}{81} = 2^6 \times \frac{3^1}{3^4} = 2^6 \times 3^{1-4} = 2^6 \times 3^{-3} \]

### Step 4: Apply the cube root
Now we can take the cube root of the simplified fraction:
\[ \sqrt[3]{\frac{192}{81}} = \sqrt[3]{2^6 \times 3^{-3}} = \sqrt[3]{2^6} \times \sqrt[3]{3^{-3}} \]

### Step 5: Simplify the cube roots
Using the property of exponents:
\[ \sqrt[3]{2^6} = 2^{6/3} = 2^2 = 4 \]
\[ \sqrt[3]{3^{-3}} = 3^{-3/3} = 3^{-1} = \frac{1}{3} \]

### Step 6: Combine the results
Now we combine the results:
\[ \sqrt[3]{\frac{192}{81}} = 4 \times \frac{1}{3} = \frac{4}{3} \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{192}{81}} \) is \( \frac{4}{3} \).

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