If √625 = 25; then√(.00000625/25)is: A. 0.0025 B. 0.001 C. 0.0001
D
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.
| Calculation Step | Value |
|---|---|
| Expression | \( \sqrt{\frac{0.00000625}{25}} \) |
| Numerator as fraction | \( \frac{625}{10^8} \) |
| Fraction inside root | \( \frac{625/10^8}{25} = \frac{25}{10^8} = 0.00000025 \) |
| Square root of fraction | \( \sqrt{0.00000025} \) |
| Final Result | \( 0.0005 \) |
Revision Table: Key Concepts for Square Roots of Decimals
| Concept | Explanation | Example |
|---|---|---|
| Square Root Definition | The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\). | \( \sqrt{25} = 5 \) because \( 5^2 = 25 \). |
| Square Root of Powers of 10 | \( \sqrt{10^{-n}} = 10^{-n/2} \). The power is halved. | \( \sqrt{10^{-8}} = 10^{-4} \). |
| Square Root of Decimals | Count the decimal places in the original number. The square root will have half that number of decimal places. | \( \sqrt{0.0025} \): 4 decimal places. \( \sqrt{25}=5 \). Result has 4/2=2 decimal places: \( 0.05 \). |
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.
Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
