What is the square root of 810?
Understand the Problem
The question is asking for the square root of the number 810. The square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. In this case, we will calculate √810.
Answer
$3\sqrt{10} \approx 9.486$
Answer for screen readers
The square root of 810 is $ 3\sqrt{10} \approx 9.486$.
Steps to Solve
- Express the number in terms of its prime factors
First, we can factor 810 into its prime factors.
$$ 810 = 2 \times 3^4 \times 5 $$
- Identify pairs of factors for square roots
When we take the square root, we can pull out pairs of prime factors. In our case:
- For $3^4$, we can take out $3^2$ (which is 9) because $3^2 \times 3^2 = 3^4$.
- The $2$ and $5$ do not have pairs.
- Calculate the square root with the extracted factors
Using the prime factors, we find the square root:
$$ \sqrt{810} = \sqrt{2} \times \sqrt{3^4} \times \sqrt{5} = \sqrt{2} \times 3^2 \times \sqrt{5} = 3 \times \sqrt{2} \times \sqrt{5} $$
- Combine the square roots
Combine the roots of $2$ and $5$ together:
$$ \sqrt{2} \times \sqrt{5} = \sqrt{10} $$
Thus,
$$ \sqrt{810} = 3 \sqrt{10} $$
- Calculate the numerical approximation
If we need a numerical value, we can approximate:
$$ \sqrt{10} \approx 3.162 $$
So,
$$ 3 \times \sqrt{10} \approx 3 \times 3.162 = 9.486 $$
The square root of 810 is $ 3\sqrt{10} \approx 9.486$.
More Information
The square root of a number can often be simplified if you can express it in terms of its prime factors. It's helpful in various mathematical operations and can be approximated for practical calculations.
Tips
- Forgetting to simplify the square root by identifying and pairing prime factors.
- Not approximating the square root accurately when needed; ensure you're using the right values for more complex numbers.
- Confusing the process of simplifying square roots with just calculating the square root directly.
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