To simplify [tex]\(\sqrt{63}\)[/tex], you need to follow these steps:
1. Identify Perfect Square Factors:
- First, we identify the largest perfect square that is a factor of 63.
- A perfect square is a number that can be expressed as the product of an integer with itself.
- In this case, the largest perfect square factor of 63 is 9, since [tex]\(9 \times 7 = 63\)[/tex].
2. Rewrite the Square Root:
- Next, you can rewrite [tex]\(\sqrt{63}\)[/tex] using its factors: [tex]\[\sqrt{63} = \sqrt{9 \times 7}\][/tex]
3. Apply the Property of Square Roots:
- We apply the property of square roots that states [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex].
- Using this property, we can rewrite the expression as: [tex]\[\sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7}\][/tex]
4. Simplify Further:
- Now, simplify [tex]\(\sqrt{9}\)[/tex]. Since 9 is a perfect square, [tex]\(\sqrt{9} = 3\)[/tex].
- Therefore: [tex]\[\sqrt{9} \times \sqrt{7} = 3 \times \sqrt{7}\][/tex]
So, the simplified form of [tex]\(\sqrt{63}\)[/tex] is:
[tex]\[\sqrt{63} = 3\sqrt{7}\][/tex]
Thus, the correct answers to fill in the blanks are:
[tex]\[
\sqrt{(} \square \sqrt{(}
\][/tex]
In this context:
- The first [tex]\(\square\)[/tex] should be filled with "63".
- The second [tex]\(\square\)[/tex] should be filled with "9 \times 7".
- The third expression simplifies the square root further to yield "3\sqrt{7}".
