`(36)^(1/6)` is equal to

Options

A. 1
B. 6
C. `sqrt6`
D. `root 3 6`

Answer

To solve the expression \( (36)^{1/6} \), we can follow these steps:

### Step 1: Rewrite 36 in terms of its prime factors
We know that:
\[ 36 = 6^2 \]

### Step 2: Substitute the prime factorization into the expression
Now we can substitute \( 36 \) in the original expression:
\[ (36)^{1/6} = (6^2)^{1/6} \]

### Step 3: Apply the power of a power property
Using the property of exponents which states that \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ (6^2)^{1/6} = 6^{2 \cdot \frac{1}{6}} = 6^{\frac{2}{6}} = 6^{\frac{1}{3}} \]

### Step 4: Interpret the result
The expression \( 6^{1/3} \) represents the cube root of 6:
\[ 6^{1/3} = \sqrt[3]{6} \]

Thus, the final answer is:
\[ (36)^{1/6} = \sqrt[3]{6} \]

### Summary
The value of \( (36)^{1/6} \) is \( \sqrt[3]{6} \).

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