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Simplify the expression:
[tex]\[ I = \frac{3 \sqrt{8} + 4 \sqrt{18} - \sqrt{32}}{\sqrt{50} - 3 \sqrt{2}} \][/tex]


Sagot :

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]