Example 7 Given that $$ a=0.1 $$ and $$ b=0.2 $$, put the following expressions in ascending order. - $$ a b= $$ a $$ \frac{a}{b} $$ - $$ b-a $$ - $$ \frac{a^{2}}{b^{2}} $$

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Let's calculate each expression step by step using the given values $$ a = 0.1 $$ and $$ b = 0.2 $$.

1. **Calculate $$ ab $$**:
   $$
   ab = 0.1 \times 0.2
   $$

2. **Calculate $$ \frac{a}{b} $$**:
   $$
   \frac{a}{b} = \frac{0.1}{0.2}
   $$

3. **Calculate $$ b - a $$**:
   $$
   b - a = 0.2 - 0.1
   $$

4. **Calculate $$ \frac{a^{2}}{b^{2}} $$**:
   $$
   \frac{a^{2}}{b^{2}} = \frac{(0.1)^{2}}{(0.2)^{2}}
   $$

Now, I will compute these expressions.
Calculate the value by following steps:
- step0: Calculate:
  $$0.2-0.1$$
- step1: Subtract the numbers:
  $$0.1$$
Calculate or simplify the expression $$ 0.1 / 0.2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.1}{0.2}$$
- step1: Convert the expressions:
  $$\frac{\frac{1}{10}}{\frac{1}{5}}$$
- step2: Multiply by the reciprocal:
  $$\frac{1}{10}\times 5$$
- step3: Reduce the numbers:
  $$\frac{1}{2}\times 1$$
- step4: Multiply:
  $$\frac{1}{2}$$
Calculate or simplify the expression $$ 0.1 * 0.2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.1\times 0.2$$
- step1: Multiply the numbers:
  $$0.02$$
Calculate or simplify the expression $$ (0.1)^2 / (0.2)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.1^{2}}{0.2^{2}}$$
- step1: Convert the expressions:
  $$\frac{\left(\frac{1}{10}\right)^{2}}{0.2^{2}}$$
- step2: Convert the expressions:
  $$\frac{\left(\frac{1}{10}\right)^{2}}{\left(\frac{1}{5}\right)^{2}}$$
- step3: Simplify:
  $$\left(\frac{\frac{1}{10}}{\frac{1}{5}}\right)^{2}$$
- step4: Simplify:
  $$\left(\frac{1}{2}\right)^{2}$$
- step5: Evaluate the power:
  $$\frac{1^{2}}{2^{2}}$$
- step6: Evaluate the power:
  $$\frac{1}{2^{2}}$$
- step7: Evaluate the power:
  $$\frac{1}{4}$$
Now, let's summarize the results of each expression:

1. $$ ab = 0.02 $$ (or $$ \frac{1}{50} $$)
2. $$ \frac{a}{b} = 0.5 $$ (or $$ \frac{1}{2} $$)
3. $$ b - a = 0.1 $$ (or $$ \frac{1}{10} $$)
4. $$ \frac{a^{2}}{b^{2}} = 0.25 $$ (or $$ \frac{1}{4} $$)

Now, we can put these values in ascending order:

- $$ ab = 0.02 $$
- $$ b - a = 0.1 $$
- $$ \frac{a^{2}}{b^{2}} = 0.25 $$
- $$ \frac{a}{b} = 0.5 $$

Thus, the ascending order is:
$$
ab < b - a < \frac{a^{2}}{b^{2}} < \frac{a}{b}
$$
The expressions in ascending order are: 1. $$ ab = 0.02 $$ 2. $$ b - a = 0.1 $$ 3. $$ \frac{a^{2}}{b^{2}} = 0.25 $$ 4. $$ \frac{a}{b} = 0.5 $$ So, the order from smallest to largest is: $$ ab < b - a < \frac{a^{2}}{b^{2}} < \frac{a}{b} $$

Example 7
Given that and , put the following expressions in ascending order.

a

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Example 7 Given that and , put the following expressions in ascending order. a