Certainly! Let's simplify [tex]\(\sqrt{63}\)[/tex].
First, we need to look for any square factors within 63 that we can simplify.
1. Prime Factorization: Let's start by finding the prime factorization of 63.
[tex]\[
63 \div 3 = 21
\][/tex]
[tex]\[
21 \div 3 = 7
\][/tex]
So,
[tex]\[
63 = 3 \times 3 \times 7 = 3^2 \times 7
\][/tex]
2. Simplifying the Square Root: Now that we have factored 63 into [tex]\(3^2 \times 7\)[/tex], we can use the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex].
[tex]\[
\sqrt{63} = \sqrt{3^2 \times 7} = \sqrt{3^2} \times \sqrt{7} = 3 \times \sqrt{7}
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{63}\)[/tex] is:
[tex]\[
3\sqrt{7}
\][/tex]
3. Numeric Simplification: To provide a numerical approximation, we calculate the value of [tex]\(3\sqrt{7}\)[/tex].
The approximate value of [tex]\(\sqrt{7}\)[/tex] is around 2.6457513110645906. Thus,
[tex]\[
3 \times \sqrt{7} \approx 3 \times 2.6457513110645906 \approx 7.937253933193772
\][/tex]
So the simplified form of [tex]\(\sqrt{63}\)[/tex] is [tex]\(3\sqrt{7}\)[/tex], which is approximately equal to 7.937253933193772 when calculated numerically.
Thus, the final answer is:
[tex]\[
\sqrt{63} = 3\sqrt{7} \approx 7.937253933193772
\][/tex]
