To simplify the expression [tex]\(\sqrt{63} \times \sqrt{1}\)[/tex], let's follow the steps below:
1. Recognize the properties of square roots:
- The square root of a product is the product of the square roots. Thus, [tex]\(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)[/tex].
2. Apply this property:
[tex]\[
\sqrt{63} \times \sqrt{1} = \sqrt{63 \times 1}
\][/tex]
3. Simplify within the square root:
[tex]\[
63 \times 1 = 63
\][/tex]
4. Evaluate the square root:
- We now have [tex]\(\sqrt{63}\)[/tex], which needs to be simplified further.
5. Factorize the number under the square root:
- 63 can be written as [tex]\(9 \times 7\)[/tex], since 9 is a perfect square:
[tex]\[
\sqrt{63} = \sqrt{9 \times 7}
\][/tex]
6. Use the property of square roots again:
[tex]\[
\sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7}
\][/tex]
7. Simplify the square root of the perfect square (9):
[tex]\[
\sqrt{9} = 3
\][/tex]
8. Combine the simplified term:
[tex]\[
\sqrt{63} = 3\sqrt{7}
\][/tex]
Given this simplification, the original expression [tex]\(\sqrt{63} \times \sqrt{1}\)[/tex] simplifies to [tex]\(3\sqrt{7}\)[/tex].
Converting this into a numerical result:
1. The value of [tex]\(\sqrt{7}\)[/tex] is approximately [tex]\(2.6457513110645906\)[/tex].
2. Hence, multiplying by 3 gives:
[tex]\[
3 \times 2.6457513110645906 \approx 7.937253933193772
\][/tex]
Therefore, the simplified expression evaluates to approximately [tex]\(7.937253933193772\)[/tex].
