square root of 164 simplified
Understand the Problem
The question is asking for the square root of 164 to be simplified. This involves finding the largest perfect square that divides 164 and rewriting the square root in its simplest form.
Answer
$2\sqrt{41}$
Answer for screen readers
The simplified form of the square root of 164 is $2\sqrt{41}$.
Steps to Solve
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Identify the prime factorization of 164
First, we need to determine the prime factors of 164. We start by dividing by the smallest prime number, which is 2:
$$ 164 \div 2 = 82 $$
Next, we divide 82 by 2:
$$ 82 \div 2 = 41 $$
Since 41 is a prime number, the complete prime factorization of 164 is:
$$ 164 = 2^2 \times 41 $$ -
Identify the largest perfect square
From the prime factorization, we see that $2^2$ is a perfect square. Thus, the largest perfect square that divides 164 is $2^2 = 4$. -
Rewrite the square root in simplified form
We can now express $\sqrt{164}$ using the largest perfect square:
$$ \sqrt{164} = \sqrt{4 \times 41} $$
Using the property of square roots, we simplify this to:
$$ \sqrt{164} = \sqrt{4} \times \sqrt{41} $$ -
Calculate the square root of the perfect square
Now we can find the square root of 4:
$$ \sqrt{4} = 2 $$
Thus, we have:
$$ \sqrt{164} = 2 \times \sqrt{41} $$
The simplified form of the square root of 164 is $2\sqrt{41}$.
More Information
Simplifying square roots is a common mathematical process, particularly useful in algebra. The prime factorization helps identify perfect squares, making it easier to simplify roots efficiently.
Tips
When simplifying square roots, a common mistake is to overlook the prime factorization or not recognizing the largest perfect square. Always ensure to factor completely and identify all possible perfect squares.
