simplify the square root of 52
Understand the Problem
The question is asking how to simplify the square root of 52. This involves factoring 52 into its prime components and identifying any perfect squares.
Answer
$2\sqrt{13}$
Answer for screen readers
The simplified form of the square root of 52 is $2\sqrt{13}$.
Steps to Solve
- Factor 52 into prime factors
To simplify the square root of 52, we start by finding its prime factorization. The number 52 can be factored as:
$$ 52 = 2 \times 26 $$
Then, we can factor 26 further:
$$ 26 = 2 \times 13 $$
So, the complete factorization of 52 is:
$$ 52 = 2^2 \times 13 $$
- Identify the perfect square
Next, we identify the perfect square within the factorization. In this case, $2^2$ is a perfect square.
- Apply the square root property
Using the property of square roots, we can split the square root of a product:
$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$
So we apply this to our factors:
$$ \sqrt{52} = \sqrt{2^2 \times 13} = \sqrt{2^2} \times \sqrt{13} $$
- Simplify the square root
Now we can simplify the square root of $2^2$:
$$ \sqrt{2^2} = 2 $$
So we have:
$$ \sqrt{52} = 2 \times \sqrt{13} $$
- Final Expression
Thus, the simplified form of $\sqrt{52}$ is:
$$ \sqrt{52} = 2\sqrt{13} $$
The simplified form of the square root of 52 is $2\sqrt{13}$.
More Information
Simplifying square roots is often used in various mathematical contexts, including algebra and geometry. It's important to identify perfect squares to make the simplification process easier. The number 52's prime factors reveal that while it is not a perfect square itself, it can be expressed in terms of one ($2^2$).
Tips
- Forgetting to fully factor the number could lead to incorrect simplification. Always ensure to factor down to prime components.
- Misapplying the square root property could result in errors. Remember that you can only separate square roots for multiplication, not addition or subtraction.
