`root(3)(8^(2))` is equal to :-

Options

A. `8^(4//3)`
B. `8^(3//2)`
C. `4^(2//3)`
D. `2^(2//3)`

Answer

To solve the expression \( \sqrt[3]{8^2} \), we can follow these steps:

### Step 1: Rewrite \( 8 \) in terms of its prime factors.
The number \( 8 \) can be expressed as:
\[ 8 = 2^3 \]

### Step 2: Substitute \( 8 \) in the expression.
Now, we can substitute \( 8 \) in the original expression:
\[ 8^2 = (2^3)^2 \]

### Step 3: Apply the power of a power property.
Using the property of exponents that states \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (2^3)^2 = 2^{3 \cdot 2} = 2^6 \]

### Step 4: Rewrite the cube root.
Now, we need to find the cube root of \( 2^6 \):
\[ \sqrt[3]{2^6} \]

### Step 5: Apply the property of roots.
Using the property that \( \sqrt[n]{a^m} = a^{m/n} \), we can simplify:
\[ \sqrt[3]{2^6} = 2^{6/3} = 2^2 \]

### Step 6: Calculate \( 2^2 \).
Now we can calculate \( 2^2 \):
\[ 2^2 = 4 \]

### Final Answer:
Thus, the value of \( \sqrt[3]{8^2} \) is:
\[ \boxed{4} \]

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