Solving for the Value of a Square Root Expression

The question asks us to find the numerical value of the expression $\frac{\sqrt{0.01}}{\sqrt{0.0025}}$. We need to calculate the square root of the numerator and the denominator separately, and then divide the results.

Step-by-Step Calculation

1. Calculate the square root of the numerator:

   The numerator is $\sqrt{0.01}$.

   We know that $0.01$ is equal to $\frac{1}{100}$.

   So, $\sqrt{0.01} = \sqrt{\frac{1}{100}}$.

   Using the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, we get:

   $\sqrt{\frac{1}{100}} = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10}$

   As a decimal, $\frac{1}{10} = 0.1$.

2. Calculate the square root of the denominator:

   The denominator is $\sqrt{0.0025}$.

   We know that $0.0025$ is equal to $\frac{25}{10000}$.

   So, $\sqrt{0.0025} = \sqrt{\frac{25}{10000}}$.

   Using the same property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, we get:

   $\sqrt{\frac{25}{10000}} = \frac{\sqrt{25}}{\sqrt{10000}} = \frac{5}{100}$

   As a decimal, $\frac{5}{100} = 0.05$.

3. Divide the results:

   Now we need to divide the value of the numerator by the value of the denominator:

   Value = $\frac{0.1}{0.05}$

   To make the division easier, we can multiply both the numerator and the denominator by 100 to remove the decimals:

   Value = $\frac{0.1 \times 100}{0.05 \times 100} = \frac{10}{5}$

   Value = $2$

Alternative Method using Square Root Properties

We can also solve this by first combining the terms under a single square root:

$\frac{\sqrt{0.01}}{\sqrt{0.0025}} = \sqrt{\frac{0.01}{0.0025}}$

To simplify the fraction inside the square root, multiply the numerator and denominator by 10000:

$\sqrt{\frac{0.01 \times 10000}{0.0025 \times 10000}} = \sqrt{\frac{100}{25}}$

Now, perform the division inside the square root:

$\sqrt{\frac{100}{25}} = \sqrt{4}$

Finally, calculate the square root:

$\sqrt{4} = 2$

Conclusion

Both methods show that the value of the expression $\frac{\sqrt{0.01}}{\sqrt{0.0025}}$ is 2.