Let's look at each part step-by-step.
### Part (d)
We need to simplify the expression:
[tex]\[ 2 \sqrt{2} - \sqrt{8} + 3 \sqrt{2} \][/tex]
1. First term: [tex]\( 2 \sqrt{2} \)[/tex]
2. Second term: [tex]\( \sqrt{8} \)[/tex]
[tex]\[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \][/tex]
So, [tex]\( \sqrt{8} = 2 \sqrt{2} \)[/tex].
3. Third term: [tex]\( 3 \sqrt{2} \)[/tex]
Now, substituting back into the original expression:
[tex]\[ 2 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2} \][/tex]
Combine like terms:
[tex]\[ (2 \sqrt{2} - 2 \sqrt{2}) + 3 \sqrt{2} = 0 + 3 \sqrt{2} = 3 \sqrt{2} \][/tex]
Hence, the simplified form of the expression is:
[tex]\[ 3 \sqrt{2} \][/tex]
### Part (e)
We need to simplify the expression:
[tex]\[ 7 \sqrt{54} - 3 \sqrt{18} + \sqrt{24} - \frac{3}{5} \sqrt{50} - \sqrt{6} \][/tex]
Let’s break down each term:
1. First term: [tex]\( 7 \sqrt{54} \)[/tex]
[tex]\[ \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6} \][/tex]
So, [tex]\( 7 \sqrt{54} = 7 \cdot 3 \sqrt{6} = 21 \sqrt{6} \)[/tex].
2. Second term: [tex]\( 3 \sqrt{18} \)[/tex]
[tex]\[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \][/tex]
So, [tex]\( 3 \sqrt{18} = 3 \cdot 3 \sqrt{2} = 9 \sqrt{2} \)[/tex].
3. Third term: [tex]\( \sqrt{24} \)[/tex]
[tex]\[ \sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2 \sqrt{6} \][/tex]
4. Fourth term: [tex]\( \frac{3}{5} \sqrt{50} \)[/tex]
[tex]\[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \][/tex]
So, [tex]\( \frac{3}{5} \sqrt{50} = \frac{3}{5} \cdot 5 \sqrt{2} = 3 \sqrt{2} \)[/tex].
5. Fifth term: [tex]\( \sqrt{6} \)[/tex]
Substituting these simplified forms into the original expression:
[tex]\[ 21 \sqrt{6} - 9 \sqrt{2} + 2 \sqrt{6} - 3 \sqrt{2} - \sqrt{6} \][/tex]
Combine like terms:
[tex]\[ (21 \sqrt{6} + 2 \sqrt{6} - \sqrt{6}) - (9 \sqrt{2} + 3 \sqrt{2}) \][/tex]
[tex]\[ (21 + 2 - 1) \sqrt{6} - (9 + 3) \sqrt{2} \][/tex]
[tex]\[ 22 \sqrt{6} - 12 \sqrt{2} \][/tex]
Hence, the simplified form of the expression is:
[tex]\[ -12 \sqrt{2} + 22 \sqrt{6} \][/tex]
Thus, the simplified forms of the given expressions are:
Part (d):
[tex]\[ 3 \sqrt{2} \][/tex]
Part (e):
[tex]\[ -12 \sqrt{2} + 22 \sqrt{6} \][/tex]
