Simplify the square root of 63.
Understand the Problem
The question is asking to simplify the expression involving the square root of 63. To simplify, we will break down 63 into its prime factors and look for perfect squares.
Answer
The simplified expression of $\sqrt{63}$ is $3\sqrt{7}$.
Answer for screen readers
The simplified expression of $\sqrt{63}$ is $3\sqrt{7}$.
Steps to Solve
- Identify the prime factors of 63
First, we need to break down 63 into its prime factors.
63 can be divided by 3:
$$ 63 \div 3 = 21 $$
Next, we can factor 21:
$$ 21 \div 3 = 7 $$
So, the prime factorization of 63 is:
$$ 63 = 3^2 \times 7 $$
- Apply the square root
Now we use the prime factorization to simplify the square root:
$$ \sqrt{63} = \sqrt{3^2 \times 7} $$
- Simplify the square root
Utilizing the property of square roots where $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, we can simplify:
$$ \sqrt{63} = \sqrt{3^2} \cdot \sqrt{7} $$
This leads us to:
$$ \sqrt{63} = 3 \cdot \sqrt{7} $$
- Final expression
The final simplified expression is:
$$ \sqrt{63} = 3\sqrt{7} $$
The simplified expression of $\sqrt{63}$ is $3\sqrt{7}$.
More Information
The square root of a number can often be simplified by expressing it in terms of its prime factors. Finding perfect squares among the prime factors allows for easy simplification, which is a useful skill in algebra.
Tips
- Forgetting to break down numbers completely into prime factors can lead to incorrect simplifications.
- Not applying the square root properties correctly, such as confusing the operations with addition or multiplication.
