What is the square root of 19600?
Understand the Problem
The question is asking for the square root of the number 19600. To solve this, we will identify what number multiplied by itself equals 19600.
Answer
$140$
Answer for screen readers
The square root of $19600$ is $140$.
Steps to Solve
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Identify the number to take the root of We need to find the square root of $19600$.
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Use prime factorization (optional) To help find the square root, we can break $19600$ down into its prime factors: $$ 19600 = 196 \times 100 $$ Breaking $196$ into prime factors: $$ 196 = 14 \times 14 = 2 \times 7 \times 2 \times 7 = 2^2 \times 7^2 $$ And breaking $100$: $$ 100 = 10 \times 10 = 2 \times 5 \times 2 \times 5 = 2^2 \times 5^2 $$
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Combine the factors Now we combine the factors: $$ 19600 = 2^2 \times 7^2 \times 2^2 \times 5^2 = 2^{2+2} \times 5^2 \times 7^2 = 2^4 \times 5^2 \times 7^2 $$
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Take the square root To find the square root, we take half the exponent of each prime factor: $$ \sqrt{19600} = \sqrt{2^4 \times 5^2 \times 7^2} = 2^{4/2} \times 5^{2/2} \times 7^{2/2} = 2^2 \times 5^1 \times 7^1 $$
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Calculate the final answer Now compute the value: $$ 2^2 = 4, \quad 5^1 = 5, \quad 7^1 = 7 $$ $$ 4 \times 5 \times 7 = 20 \times 7 = 140 $$
The square root of $19600$ is $140$.
More Information
The square root helps determine which number multiplied by itself gives the original number. In this case, $140 \times 140 = 19600$, demonstrating that our solution is correct.
Tips
- Forgetting to simplify factors before taking the square root. Always simplify the expression fully to avoid errors.
- Miscalculating the multiplication of the results after taking the square root. Carefully check each step.
