To solve the expression [tex]\( I_N = \frac{3 \sqrt{8} + 4 \sqrt{18} - \sqrt{32}}{\sqrt{50} - 3 \sqrt{2}} \)[/tex], follow these steps in detail:
1. Simplify the radicals:
- [tex]\( \sqrt{8} = 2 \sqrt{2} \)[/tex]
- [tex]\( \sqrt{18} = 3 \sqrt{2} \)[/tex]
- [tex]\( \sqrt{32} = 4 \sqrt{2} \)[/tex]
- [tex]\( \sqrt{50} = 5 \sqrt{2} \)[/tex]
2. Substitute these simplifications back into the expression:
[tex]\[
I_N = \frac{3 \cdot 2 \sqrt{2} + 4 \cdot 3 \sqrt{2} - 4 \sqrt{2}}{5 \sqrt{2} - 3 \sqrt{2}}
\][/tex]
3. Multiply and combine like terms in the numerator:
- [tex]\( 3 \cdot 2 \sqrt{2} = 6 \sqrt{2} \)[/tex]
- [tex]\( 4 \cdot 3 \sqrt{2} = 12 \sqrt{2} \)[/tex]
- [tex]\( -4 \sqrt{2} \)[/tex]
Combine these terms:
[tex]\[
6 \sqrt{2} + 12 \sqrt{2} - 4 \sqrt{2} = (6 + 12 - 4) \sqrt{2} = 14 \sqrt{2}
\][/tex]
4. Simplify the denominator:
[tex]\[
5 \sqrt{2} - 3 \sqrt{2} = 2 \sqrt{2}
\][/tex]
5. Rewrite the expression with the simplified terms:
[tex]\[
I_N = \frac{14 \sqrt{2}}{2 \sqrt{2}}
\][/tex]
6. Simplify the division:
Since both numerator and denominator have [tex]\( \sqrt{2} \)[/tex], they cancel out:
[tex]\[
I_N = \frac{14}{2} = 7
\][/tex]
So, the final answer is:
[tex]\[ I_N = 7 \][/tex]
