To simplify the expression [tex]\(\frac{12}{\sqrt{2}}+\sqrt{18}\)[/tex] into the form [tex]\(b \sqrt{2}\)[/tex], where [tex]\(b\)[/tex] is an integer, we can follow these steps:
1. Rationalize the denominator of [tex]\(\frac{12}{\sqrt{2}}\)[/tex]:
To simplify [tex]\(\frac{12}{\sqrt{2}}\)[/tex], we multiply both the numerator and denominator by [tex]\(\sqrt{2}\)[/tex] to rationalize the denominator:
[tex]\[
\frac{12}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}
\][/tex]
2. Simplify [tex]\(\sqrt{18}\)[/tex]:
The square root of 18 can be broken down into its prime factors:
[tex]\[
\sqrt{18} = \sqrt{2 \times 9} = \sqrt{2} \times \sqrt{9} = \sqrt{2} \times 3 = 3\sqrt{2}
\][/tex]
3. Add the simplified terms together:
Combining the two parts we have:
[tex]\[
6\sqrt{2} + 3\sqrt{2}
\][/tex]
Since both terms have [tex]\(\sqrt{2}\)[/tex] as a common factor, we can add the coefficients:
[tex]\[
6\sqrt{2} + 3\sqrt{2} = (6 + 3)\sqrt{2} = 9\sqrt{2}
\][/tex]
Hence, the expression [tex]\(\frac{12}{\sqrt{2}}+\sqrt{18}\)[/tex] simplifies to [tex]\(9\sqrt{2}\)[/tex]. Therefore, [tex]\(b\)[/tex] is [tex]\(9\)[/tex].
