To determine whether each of the given variables represents a rational or irrational number, follow these steps:
### Step 1: Solve for the values
(i) [tex]\( x^2 = 5 \)[/tex]
First, solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{5} \][/tex]
The square root of 5 is approximately 2.23606797749979.
(ii) [tex]\( y^2 = 9 \)[/tex]
Next, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt{9} \][/tex]
The square root of 9 is exactly 3.0.
(iii) [tex]\( z^2 = 0.04 \)[/tex]
Then, solve for [tex]\( z \)[/tex]:
[tex]\[ z = \sqrt{0.04} \][/tex]
The square root of 0.04 is 0.2.
(iv) [tex]\( u^2 = \frac{17}{4} \)[/tex]
Finally, solve for [tex]\( u \)[/tex]:
[tex]\[ u = \sqrt{\frac{17}{4}} \][/tex]
The square root of [tex]\( \frac{17}{4} \)[/tex] is approximately 2.0615528128088303.
### Step 2: Determine Rationality or Irrationality
(i) [tex]\( x = \sqrt{5} \approx 2.23606797749979 \)[/tex]
The value of the square root of 5 is not a rational number since it cannot be expressed as a simple fraction. Therefore, [tex]\( x \)[/tex] is irrational.
(ii) [tex]\( y = \sqrt{9} = 3.0 \)[/tex]
The value of the square root of 9 is a whole number (3). Since whole numbers are rational, [tex]\( y \)[/tex] is rational.
(iii) [tex]\( z = \sqrt{0.04} = 0.2 \)[/tex]
The value of the square root of 0.04 is 0.2, which can be expressed as a fraction (1/5). Since it can be written as a fraction with integer numerator and non-zero integer denominator, [tex]\( z \)[/tex] is rational.
(iv) [tex]\( u = \sqrt{\frac{17}{4}} \approx 2.0615528128088303 \)[/tex]
The value of the square root of [tex]\( \frac{17}{4} \)[/tex] is not a rational number since it cannot be expressed as a simple fraction. Therefore, [tex]\( u \)[/tex] is irrational.
### Conclusion
Thus, the classifications are as follows:
- [tex]\( x \)[/tex] is irrational.
- [tex]\( y \)[/tex] is rational.
- [tex]\( z \)[/tex] is rational.
- [tex]\( u \)[/tex] is irrational.
