What is the square root of \(\frac{{{{\left( {0.35} \right)}^2} + {\rm{\;}}0.70{\rm{\;}} + {\rm{\;}}1}}{{2.25}} + 0.19{\rm{\;}}?\)
1
Understanding the Square Root Problem
The question asks us to find the square root of a complex expression involving decimals. We need to simplify the expression inside the square root first and then calculate the square root of the resulting value. The expression is \(\frac{{{{\left( {0.35} \right)}^2} + {\rm{\;}}0.70{\rm{\;}} + {\rm{\;}}1}}{{2.25}} + 0.19{\rm{\;}}\).
Step-by-Step Solution to Find the Square Root
Let's break down the calculation step by step:
- Simplify the numerator: The numerator is \({\left( {0.35} \right)^2} + {\rm{\;}}0.70{\rm{\;}} + {\rm{\;}}1\).
We can rewrite \(0.70\) as \(2 \times 0.35 \times 1\) and \(1\) as \(1^2\). The numerator becomes \({\left( {0.35} \right)^2} + 2 \times 0.35 \times 1 + {\left( 1 \right)^2}\).
This expression is in the form of \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = 0.35\) and \(b = 1\).
So, the numerator simplifies to \({\left( {0.35 + 1} \right)^2} = {\left( {1.35} \right)^2}\).
- Simplify the denominator: The denominator is \(2.25\).
We can recognize that \(2.25\) is the square of \(1.5\) (since \(1.5 \times 1.5 = 2.25\)).
So, the denominator can be written as \({\left( {1.5} \right)^2}\).
- Simplify the fraction: The fraction is \(\frac{{{{\left( {1.35} \right)}^2}}}{{{{\left( {1.5} \right)}^2}}}\).
Using the property \({\left( {\frac{a}{b}} \right)^m} = \frac{{{a^m}}}{{{b^m}}}\), we can write this as \({\left( {\frac{{1.35}}{{1.5}}} \right)^2}\).
Now, let's simplify the fraction \(\frac{{1.35}}{{1.5}}\). We can multiply the numerator and denominator by 100 to remove decimals: \(\frac{{1.35 \times 100}}{{1.5 \times 100}} = \frac{{135}}{{150}}\).
Divide both the numerator and denominator by their greatest common divisor, which is 15:
\(\frac{{135 \div 15}}{{150 \div 15}} = \frac{9}{{10}} = 0.9\).
So, the fraction simplifies to \({\left( {0.9} \right)^2} = 0.81\).
- Add the remaining term: The expression inside the square root is \(\frac{{{{\left( {0.35} \right)}^2} + {\rm{\;}}0.70{\rm{\;}} + {\rm{\;}}1}}{{2.25}} + 0.19\).
Substituting the simplified fraction, we get \(0.81 + 0.19\).
- Calculate the sum: \(0.81 + 0.19 = 1.00 = 1\).
- Calculate the square root: The question asks for the square root of the entire expression, which we found to be 1.
The square root is \(\sqrt{1}\).
\(\sqrt{1} = 1\).
The final result is 1.
| Calculation Step | Expression | Result |
|---|---|---|
| Numerator simplification | \({\left( {0.35} \right)^2} + {\rm{\;}}0.70{\rm{\;}} + {\rm{\;}}1\) | \({\left( {1.35} \right)^2}\) |
| Denominator simplification | \(2.25\) | \({\left( {1.5} \right)^2}\) |
| Fraction simplification | \(\frac{{{{\left( {1.35} \right)}^2}}}{{{{\left( {1.5} \right)}^2}}}\) | \(0.81\) |
| Total inside square root | \(0.81 + 0.19\) | \(1\) |
| Final square root | \(\sqrt{1}\) | \(1\) |
Comparing our result with the given options, we find that the value is 1.
Revision Table: Key Concepts for Square Root Problems
| Concept | Description | Formula/Example |
|---|---|---|
| Square Root | A value that, when multiplied by itself, gives the number. Represented by the symbol \(\sqrt{\;}\). | \(\sqrt{x} = y\) means \(y^2 = x\). Example: \(\sqrt{9} = 3\) because \(3^2 = 9\). |
| Squaring Decimals | Multiplying a decimal number by itself. | \((0.5)^2 = 0.5 \times 0.5 = 0.25\). |
| Algebraic Identity | Formula useful for simplifying expressions. | \((a+b)^2 = a^2 + 2ab + b^2\). Used to simplify the numerator in this problem. |
| Simplifying Fractions | Reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor. Also involves handling decimals. | \(\frac{135}{150} = \frac{135 \div 15}{150 \div 15} = \frac{9}{10}\). |
Additional Information on Simplifying Expressions
When faced with complex expressions involving square roots, fractions, and decimals, it's crucial to follow the order of operations (PEMDAS/BODMAS) and simplify the expression inside the root first. Recognizing algebraic identities can significantly speed up the simplification process, as seen in the numerator of this problem.
Handling decimals in fractions often involves converting them to whole numbers by multiplying by powers of 10, making the division or simplification easier. Always check if the denominator is a perfect square, which might offer another way to simplify the fraction under the square root.
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