Find the value of `{(216)^(2/3)+(36)^(-1/2)}`

Options

A. `(216)/(6)`
B. `(218)/(6)`
C. `(215)/(6)`
D. `(217)/(6)`

Answer

To find the value of \( (216)^{2/3} + (36)^{-1/2} \), we will break down the calculations step by step.

### Step 1: Calculate \( (216)^{2/3} \)

To simplify \( (216)^{2/3} \), we can first find \( (216)^{1/3} \) and then square the result.

1. **Find the cube root of 216**:
\[ 216 = 6^3 \implies (216)^{1/3} = 6 \]

2. **Now square the result**:
\[ (216)^{2/3} = (6)^2 = 36 \]

### Step 2: Calculate \( (36)^{-1/2} \)

Next, we simplify \( (36)^{-1/2} \).

1. **Find the square root of 36**:
\[ 36 = 6^2 \implies (36)^{1/2} = 6 \]

2. **Apply the negative exponent**:
\[ (36)^{-1/2} = \frac{1}{(36)^{1/2}} = \frac{1}{6} \]

### Step 3: Combine the results

Now we can add the results from Step 1 and Step 2:
\[ (216)^{2/3} + (36)^{-1/2} = 36 + \frac{1}{6} \]

### Step 4: Convert to a common denominator

To add \( 36 \) and \( \frac{1}{6} \), we convert \( 36 \) into a fraction with a denominator of 6:
\[ 36 = \frac{36 \times 6}{6} = \frac{216}{6} \]

Now we can add:
\[ \frac{216}{6} + \frac{1}{6} = \frac{216 + 1}{6} = \frac{217}{6} \]

### Final Answer

Thus, the final value is:
\[ \frac{217}{6} \]

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