Which of the following numbers is irrational?

Options

A. `root(10)(1024)`
B. `root(4)(1024)`
C. `sqrt(1024)`
D. `root(5)(1024)`

Answer

To determine which of the following numbers is irrational, we will analyze each option step by step.

### Step 1: Understanding the Number 1024
First, we recognize that 1024 can be expressed as a power of 2:
\[ 1024 = 2^{10} \]

### Step 2: Analyze Each Option
Now, we will check each option one by one to see if the result is rational or irrational.

#### Option 1: \( 1024^{1/10} \)
Using the expression for 1024:
\[ 1024^{1/10} = (2^{10})^{1/10} = 2^{10 \cdot \frac{1}{10}} = 2^1 = 2 \]
**Conclusion**: This is a rational number.

#### Option 2: \( 1024^{1/4} \)
Using the expression for 1024:
\[ 1024^{1/4} = (2^{10})^{1/4} = 2^{10 \cdot \frac{1}{4}} = 2^{2.5} = 2^{2 + 0.5} = 2^2 \cdot 2^{0.5} = 4 \sqrt{2} \]
**Conclusion**: Since \( \sqrt{2} \) is irrational, \( 4\sqrt{2} \) is also irrational.

#### Option 3: \( \sqrt{1024} \)
This can be simplified as:
\[ \sqrt{1024} = (1024)^{1/2} = (2^{10})^{1/2} = 2^{10 \cdot \frac{1}{2}} = 2^5 = 32 \]
**Conclusion**: This is a rational number.

#### Option 4: \( 1024^{1/5} \)
Using the expression for 1024:
\[ 1024^{1/5} = (2^{10})^{1/5} = 2^{10 \cdot \frac{1}{5}} = 2^{2} = 4 \]
**Conclusion**: This is a rational number.

### Final Conclusion
Among the options analyzed, the only irrational number is:
\[ \text{Option 2: } 1024^{1/4} = 4\sqrt{2} \]