Solving the Square Root Expression

We are asked to find the value of the expression \( \sqrt{\frac{0.00000625}{25}} \), given that \( \sqrt{625} = 25 \).

Let's break down the problem step-by-step:

Step 1: Simplify the fraction inside the square root

The expression inside the square root is \( \frac{0.00000625}{25} \).

We can rewrite the numerator \( 0.00000625 \) in scientific notation or as a fraction to make the division easier. Note that \( 0.00000625 \) has 8 decimal places.

\( 0.00000625 = \frac{625}{100,000,000} = \frac{625}{10^8} \)

Now, the division becomes:

\( \frac{0.00000625}{25} = \frac{\frac{625}{10^8}}{25} = \frac{625}{10^8 \times 25} \)

We know that \( \frac{625}{25} = 25 \). So, the expression simplifies to:

\( \frac{25}{10^8} \)

In decimal form, \( \frac{25}{10^8} = 25 \times 10^{-8} = 0.00000025 \).

So, the expression inside the square root is \( 0.00000025 \).

Step 2: Calculate the square root of the simplified value

Now we need to find \( \sqrt{0.00000025} \).

We know that \( \sqrt{25} = 5 \).

The number \( 0.00000025 \) has 8 decimal places. When we take the square root of a number with a certain number of decimal places, the result will have half that number of decimal places.

Half of 8 is 4.

So, \( \sqrt{0.00000025} \) will be a number with the digit 5 and 4 decimal places.

This number is \( 0.0005 \).

Alternatively, using scientific notation:

\( \sqrt{0.00000025} = \sqrt{25 \times 10^{-8}} \)

Using the property \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \):

\( \sqrt{25 \times 10^{-8}} = \sqrt{25} \times \sqrt{10^{-8}} \)

We know \( \sqrt{25} = 5 \).

And \( \sqrt{10^{-8}} = (10^{-8})^{\frac{1}{2}} = 10^{-8 \times \frac{1}{2}} = 10^{-4} \).

So, the result is \( 5 \times 10^{-4} \).

Converting this back to decimal form: \( 5 \times 10^{-4} = 0.0005 \).

Thus, \( \sqrt{\frac{0.00000625}{25}} = 0.0005 \).

Checking the Options

• A. 0.0025
• B. 0.001
• C. 0.0001
• D. 0.0005

Our calculated value \( 0.0005 \) matches option D.

Revision Table: Key Concepts for Square Roots of Decimals

Additional Information: Handling Decimal Division

Dividing decimal numbers can sometimes be tricky. A simple method is to remove the decimal places by multiplying both the numerator and the denominator by a power of 10. In our case, we divided \(0.00000625\) by \(25\).

\( \frac{0.00000625}{25} \)

To remove the decimal from the numerator, multiply both numerator and denominator by \(10^8\):

\( \frac{0.00000625 \times 10^8}{25 \times 10^8} = \frac{625}{25 \times 10^8} \)

Now, perform the integer division \( \frac{625}{25} = 25 \).

So, the expression becomes \( \frac{25}{10^8} \). This is the same result obtained earlier, \( 0.00000025 \).

Understanding how to manipulate decimals and powers of 10 is crucial for solving such problems efficiently.