If `x = (0.25) ^((1)/(2)) , y = (0.4) ^(2), z = ( 0. 216) ^((1)/(3)) ` then

Options

A. `Y gt X gt Z`
B. `X gt Y gt Z`
C. `Z gt X gt Y`
D. `X gt Z gt Y`

Answer

To solve the problem, we need to evaluate the expressions for \( x \), \( y \), and \( z \) step by step.

### Step 1: Calculate \( x = (0.25)^{(1/2)} \)

1. Rewrite \( 0.25 \) as a fraction:
\[ 0.25 = \frac{25}{100} = \frac{1}{4} \]
2. Now, take the square root:
\[ x = \left(\frac{1}{4}\right)^{(1/2)} = \frac{1^{(1/2)}}{4^{(1/2)}} = \frac{1}{2} = 0.5 \]

### Step 2: Calculate \( y = (0.4)^2 \)

1. Rewrite \( 0.4 \) as a fraction:
\[ 0.4 = \frac{4}{10} = \frac{2}{5} \]
2. Now, square the fraction:
\[ y = \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} = 0.16 \]

### Step 3: Calculate \( z = (0.216)^{(1/3)} \)

1. Rewrite \( 0.216 \) as a fraction:
\[ 0.216 = \frac{216}{1000} \]
2. Now, take the cube root:
\[ z = \left(\frac{216}{1000}\right)^{(1/3)} = \frac{216^{(1/3)}}{1000^{(1/3)}} \]
3. Calculate the cube roots:
\[ 216^{(1/3)} = 6 \quad \text{and} \quad 1000^{(1/3)} = 10 \]
Therefore,
\[ z = \frac{6}{10} = 0.6 \]

### Step 4: Compare \( x \), \( y \), and \( z \)

Now we have:
- \( x = 0.5 \)
- \( y = 0.16 \)
- \( z = 0.6 \)

We can now compare these values:
- \( z > x > y \)
- Thus, the correct order is \( z > x > y \).

### Conclusion

The correct option is \( z > x > y \).

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