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 question, we need to calculate the values of \( x \), \( y \), and \( z \) as defined in the question, and then compare them. Let's break it down step by step.

### Step 1: Calculate \( x \)

Given:
\[ x = (0.25)^{\frac{1}{2}} \]

We can rewrite \( 0.25 \) as a fraction:
\[ 0.25 = \frac{25}{100} = \frac{1}{4} \]

Now, we can express \( x \) as:
\[ x = \left(\frac{1}{4}\right)^{\frac{1}{2}} = \frac{1^{\frac{1}{2}}}{4^{\frac{1}{2}}} = \frac{1}{\sqrt{4}} = \frac{1}{2} = 0.5 \]

### Step 2: Calculate \( y \)

Given:
\[ y = (0.4)^2 \]

We can rewrite \( 0.4 \) as a fraction:
\[ 0.4 = \frac{4}{10} = \frac{2}{5} \]

Now, we can express \( y \) as:
\[ y = \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} = 0.16 \]

### Step 3: Calculate \( z \)

Given:
\[ z = (0.216)^{\frac{1}{3}} \]

We can rewrite \( 0.216 \) as a fraction:
\[ 0.216 = \frac{216}{1000} \]

Now, we can express \( z \) as:
\[ z = \left(\frac{216}{1000}\right)^{\frac{1}{3}} = \frac{216^{\frac{1}{3}}}{1000^{\frac{1}{3}}} \]

Calculating \( 216^{\frac{1}{3}} \):
\[ 216 = 6^3 \implies 216^{\frac{1}{3}} = 6 \]

Calculating \( 1000^{\frac{1}{3}} \):
\[ 1000 = 10^3 \implies 1000^{\frac{1}{3}} = 10 \]

Thus:
\[ z = \frac{6}{10} = 0.6 \]

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

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

To compare these values:
- \( y < x < z \)
- Therefore, the order from smallest to largest is \( y < x < z \).

### Final Answer

The smallest value is \( y = 0.16 \), followed by \( x = 0.5 \), and the largest is \( z = 0.6 \).

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