Problem 47
Question
\(\sqrt{49 \mathrm{~km}^{2}}\)
Step-by-Step Solution
Verified Answer
7 km
1Step 1 - Identify the Expression
The given expression is \(\backslash\backslashsqrt{49 \mathrm{~km}^{2}}\). This represents the square root of 49 square kilometers.
2Step 2 - Simplify the Numerical Value
Calculate the square root of the numerical value first. The square root of 49 is 7.
3Step 3 - Simplify the Unit
Since taking the square root of \(\backslash\backslashmathrm{~km}^2\) removes the exponent, the unit simplifies to kilometers (km).
4Step 4 - Combine the Results
Combine the simplified numerical value from Step 2 with the simplified unit from Step 3. Therefore, \(\backslash\backslashsqrt{49 \mathrm{~km}^{2}} = 7 \mathrm{~km}\).
Key Concepts
Square RootsUnit ConversionSimplifying Algebraic Expressions
Square Roots
Square roots are a fundamental mathematical concept that simplify expressions involving exponents. They represent a value that, when multiplied by itself, gives the original number. For example, the square root of 49 is 7 because 7 multiplied by 7 equals 49.
Square roots are often represented using the radical symbol \( \sqrt{ } \). This symbol indicates that you need to find the number that, when squared, produces the given value. For instance, \( \sqrt{49} = 7 \). Square roots can be found for both perfect squares (like 49) and non-perfect squares (like 50), although non-perfect squares result in irrational numbers.
Square roots are often represented using the radical symbol \( \sqrt{ } \). This symbol indicates that you need to find the number that, when squared, produces the given value. For instance, \( \sqrt{49} = 7 \). Square roots can be found for both perfect squares (like 49) and non-perfect squares (like 50), although non-perfect squares result in irrational numbers.
Unit Conversion
Unit conversion is an essential skill in both mathematics and science. It involves changing one unit of measurement to another, making calculations more straightforward and understandable. In the given problem, the unit is square kilometers (km²), and we need to convert it to kilometers (km).
When dealing with square roots of units, the process is similar to numeric values. The square root of \( km^2 \) simplifies to km. This is because taking the square root of \( km^2 \) removes the exponent, just as you would remove the power of two in a numeric context. Therefore, \( \sqrt{49 \mathrm{~km}^2} = 7 \mathrm{~km} \). Unit conversion ensures that our results are in understandable and practical units.
When dealing with square roots of units, the process is similar to numeric values. The square root of \( km^2 \) simplifies to km. This is because taking the square root of \( km^2 \) removes the exponent, just as you would remove the power of two in a numeric context. Therefore, \( \sqrt{49 \mathrm{~km}^2} = 7 \mathrm{~km} \). Unit conversion ensures that our results are in understandable and practical units.
Simplifying Algebraic Expressions
Simplifying algebraic expressions is about reducing complexity while keeping the expression equivalent. It involves combining like terms, solving for unknowns, and breaking down complicated expressions into more manageable parts.
In the given exercise, the expression \( \sqrt{49 \mathrm{~km}^2} \) is simplified in a few clear steps. First, you identify the expression. Next, you simplify the numerical value, which is straightforward as the square root of 49 is 7. Then, you tackle the unit, reducing \( km^2 \) to km by removing the exponent through the square root operation. Finally, you combine these results to get 7 km. Simplifying expressions like this helps in solving more complex problems and understanding underlying patterns.
In the given exercise, the expression \( \sqrt{49 \mathrm{~km}^2} \) is simplified in a few clear steps. First, you identify the expression. Next, you simplify the numerical value, which is straightforward as the square root of 49 is 7. Then, you tackle the unit, reducing \( km^2 \) to km by removing the exponent through the square root operation. Finally, you combine these results to get 7 km. Simplifying expressions like this helps in solving more complex problems and understanding underlying patterns.
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