What is the square root of 800?
Understand the Problem
The question is asking for the square root of 800, which is a mathematical operation that requires calculating the number that, when multiplied by itself, equals 800.
Answer
The square root of 800 is $20\sqrt{2}$.
Answer for screen readers
The square root of 800 is $20\sqrt{2}$.
Steps to Solve
- Express 800 as a Product of its Prime Factors
To simplify the square root, we first break down 800 into its prime factors.
$$ 800 = 8 \times 100 $$
Next, we factor each component:
$$ 8 = 2^3 $$ $$ 100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2 $$
Putting these together:
$$ 800 = 2^3 \times (2^2 \times 5^2) = 2^{3+2} \times 5^2 = 2^5 \times 5^2 $$
- Use the Square Root Property
Now, we can apply the square root to both sides:
$$ \sqrt{800} = \sqrt{2^5 \times 5^2} $$
Using the property of square roots, $ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$:
$$ \sqrt{800} = \sqrt{2^5} \times \sqrt{5^2} $$
- Calculate Each Square Root
To find these square roots, we simplify:
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For $ \sqrt{5^2} $, it simplifies to $ 5 $.
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For $ \sqrt{2^5} $, we can express it as:
$$ \sqrt{2^5} = \sqrt{(2^4) \times 2} = \sqrt{2^4} \times \sqrt{2} = 4\sqrt{2} $$
- Combine the Results
Now we can combine the results to find:
$$ \sqrt{800} = 4\sqrt{2} \times 5 = 20\sqrt{2} $$
The square root of 800 is $20\sqrt{2}$.
More Information
The square root of 800, $20\sqrt{2}$, can also be approximated as approximately $28.28$ when using a calculator. This shows how square roots can be expressed in both exact and approximate forms.
Tips
- Forgetting to break down the number into its prime factors can lead to a more complicated calculation.
- Neglecting the square root properties when combining terms might cause mistakes in simplification. Always remember $ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $.
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