Sure, let's simplify the given surd expressions step-by-step.
### Part (a)
[tex]\[ 7 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} - 3 \sqrt{3} \][/tex]
1. Identify and combine like terms. All the terms are multiples of [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ (7 - 2 + 1 - 3) \sqrt{3} \][/tex]
2. Simplify the constants inside the parentheses:
[tex]\[ (7 - 2 + 1 - 3) = 3 \][/tex]
3. Multiply the simplified constant by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ 3 \sqrt{3} \][/tex]
Thus, the simplified form of the expression [tex]\(7 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} - 3 \sqrt{3}\)[/tex] is:
[tex]\[ 3 \sqrt{3} \][/tex]
Evaluating this numerically:
[tex]\[ 3\sqrt{3} \approx 5.196 \][/tex]
### Part (b)
[tex]\[ 5 \sqrt{7} + 4 \sqrt{7} - 8 \sqrt{7} \][/tex]
1. Similarly, identify and combine the like terms, which are multiples of [tex]\(\sqrt{7}\)[/tex]:
[tex]\[ (5 + 4 - 8) \sqrt{7} \][/tex]
2. Simplify the constants inside the parentheses:
[tex]\[ (5 + 4 - 8) = 1 \][/tex]
3. Multiply the simplified constant by [tex]\(\sqrt{7}\)[/tex]:
[tex]\[ 1 \sqrt{7} \][/tex]
Thus, the simplified form of the expression [tex]\(5 \sqrt{7} + 4 \sqrt{7} - 8 \sqrt{7}\)[/tex] is:
[tex]\[ \sqrt{7} \][/tex]
Evaluating this numerically:
[tex]\[ \sqrt{7} \approx 2.646 \][/tex]
### Final Simplified Forms are:
a) [tex]\(3 \sqrt{3} \approx 5.196\)[/tex]
b) [tex]\(\sqrt{7} \approx 2.646\)[/tex]
