State, with reason which of the are surds and which are not :
`root(4)27`

Answer

To determine whether the expression \( \sqrt[4]{27} \) is a surd or not, we will follow these steps:

### Step 1: Understand the Definition of a Surd
A surd is defined as an irrational number that can be expressed in the form of a root. Specifically, it is a root of a positive integer that cannot be simplified to a rational number.

### Step 2: Analyze the Given Expression
The expression we have is \( \sqrt[4]{27} \). This represents the fourth root of 27.

### Step 3: Simplify the Expression
To determine if \( \sqrt[4]{27} \) is a surd, we need to check if it can be simplified to a rational number.

1. **Finding the prime factorization of 27:**
\[ 27 = 3^3 \]

2. **Expressing the fourth root:**
\[ \sqrt[4]{27} = \sqrt[4]{3^3} \]

3. **Using the property of exponents:**
\[ \sqrt[4]{3^3} = 3^{3/4} \]

### Step 4: Determine if the Result is Rational or Irrational
The exponent \( \frac{3}{4} \) indicates that \( 3^{3/4} \) is not a whole number, meaning \( \sqrt[4]{27} \) cannot be expressed as a simple fraction. Therefore, it is an irrational number.

### Conclusion
Since \( \sqrt[4]{27} \) is an irrational number that cannot be simplified to a rational number, we conclude that:

- **\( \sqrt[4]{27} \) is a surd.**