Simplify the expression \((\sqrt{81} + \sqrt{0.81} + \sqrt{0.0081})\sqrt{10000}.\)
Simplifying Expressions with Square Roots: A Step-by-Step Guide
Let's simplify the given mathematical expression step by step. The expression is \((\sqrt{81} + \sqrt{0.81} + \sqrt{0.0081})\sqrt{10000}.\) To simplify this, we need to calculate the value of each square root term first, then perform the addition inside the parentheses, and finally multiply the result by the value of the last square root.
Step 1: Calculate each square root
We have three square roots inside the parentheses and one outside. Let's find the value of each one:
- Calculating \(\sqrt{81}\): The square root of 81 is the number that, when multiplied by itself, gives 81. This number is 9.
\(\sqrt{81} = 9\)
- Calculating \(\sqrt{0.81}\): The number 0.81 can be written as a fraction \(\frac{81}{100}\). The square root of a fraction is the square root of the numerator divided by the square root of the denominator.
\(\sqrt{0.81} = \sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} = \frac{9}{10} = 0.9\)
- Calculating \(\sqrt{0.0081}\): The number 0.0081 can be written as a fraction \(\frac{81}{10000}\). Similar to the previous step, we take the square root of the numerator and the denominator.
\(\sqrt{0.0081} = \sqrt{\frac{81}{10000}} = \frac{\sqrt{81}}{\sqrt{10000}} = \frac{9}{100} = 0.09\)
- Calculating \(\sqrt{10000}\): The square root of 10000 is the number that, when multiplied by itself, gives 10000. This number is 100.
\(\sqrt{10000} = 100\)
Step 2: Substitute the values back into the expression
Now we replace the square roots in the original expression with their calculated values:
\((\sqrt{81} + \sqrt{0.81} + \sqrt{0.0081})\sqrt{10000} = (9 + 0.9 + 0.09) \times 100\)
Step 3: Perform the addition inside the parentheses
Add the decimal numbers inside the parentheses:
\(9 + 0.9 + 0.09 = 9.99\)
So the expression becomes:
\((9.99) \times 100\)
Step 4: Perform the multiplication
Finally, multiply the sum by 100:
\(9.99 \times 100 = 999\)
Final Result of Simplifying the Expression
The simplified value of the expression \((\sqrt{81} + \sqrt{0.81} + \sqrt{0.0081})\sqrt{10000}\) is 999.
| Square Root | Value |
|---|---|
| \(\sqrt{81}\) | 9 |
| \(\sqrt{0.81}\) | 0.9 |
| \(\sqrt{0.0081}\) | 0.09 |
| \(\sqrt{10000}\) | 100 |
Revision Table: Key Concepts for Simplifying Square Roots
| Concept | Explanation | Example |
|---|---|---|
| Square Root | A number that produces a given quantity when multiplied by itself. | \(\sqrt{25} = 5\) because \(5 \times 5 = 25\) |
| Square Root of Decimals | Convert decimal to fraction, find square roots of numerator and denominator. | \(\sqrt{0.04} = \sqrt{\frac{4}{100}} = \frac{\sqrt{4}}{\sqrt{100}} = \frac{2}{10} = 0.2\) |
| Square Root of Powers of 10 | \(\sqrt{10^n} = 10^{n/2}\). For even \(n\), the result is an integer power of 10. | \(\sqrt{100} = \sqrt{10^2} = 10^{2/2} = 10^1 = 10\), \(\sqrt{10000} = \sqrt{10^4} = 10^{4/2} = 10^2 = 100\) |
Additional Information: Understanding Decimal Square Roots
When calculating the square root of a decimal number, the number of decimal places in the result is half the number of decimal places in the original number. For example:
- 0.81 has 2 decimal places. \(\sqrt{0.81} = 0.9\), which has 1 decimal place (2/2 = 1).
- 0.0081 has 4 decimal places. \(\sqrt{0.0081} = 0.09\), which has 2 decimal places (4/2 = 2).
- If you had \(\sqrt{0.000081}\), it has 6 decimal places. The result would have 6/2 = 3 decimal places: 0.009.
This rule helps you quickly check if your decimal square root calculation is likely correct based on the number of decimal places.
Important Questions from Decimals
1254 + 125.4 + 12.54 + 1.254 = ?
The value of 1/0.24 of 1.44 is:
The value of 80.6 ÷ 4030 = ?
The decimal expansion of \(\frac{31}{2.5}\) will terminate after:
- The decimal expression of \(\frac{3}{8}\) comes to an end after how many digits after the decimal?
