Sure, let's simplify each of the given expressions step-by-step.
1. [tex]\( (\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) \)[/tex]:
Let's use the distributive property (multiplying each term inside the first parenthesis by each term inside the second parenthesis):
[tex]\[
(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) = \sqrt{2} \cdot \sqrt{5} + \sqrt{2} \cdot (-\sqrt{7}) + \sqrt{3} \cdot \sqrt{5} + \sqrt{3} \cdot (-\sqrt{7})
\][/tex]
Simplify each term:
[tex]\[
= \sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21} \approx -1.2889720753539853
\][/tex]
2. [tex]\( \sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21} \)[/tex]:
This expression is already in its simplest form but we can group and balance the terms:
[tex]\[
= \sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21} \approx -1.288972075353985
\][/tex]
3. [tex]\( 2 \sqrt{5} - 2 \sqrt{7} \)[/tex]:
Factor out the common coefficient [tex]\( 2 \)[/tex]:
[tex]\[
= 2 (\sqrt{5} - \sqrt{7})
\][/tex]
The expression simplifies to:
[tex]\[
2 (\sqrt{5} - \sqrt{7}) \approx -0.8193666671296018
\][/tex]
4. [tex]\( 2 \sqrt{5} + 3 \sqrt{5} - 2 \sqrt{7} - 3 \sqrt{7} \)[/tex]:
Group the terms with common radicals:
[tex]\[
= (2 \sqrt{5} + 3 \sqrt{5}) - (2 \sqrt{7} + 3 \sqrt{7})
\][/tex]
Simplify inside each parenthesis:
[tex]\[
= 5 \sqrt{5} - 5 \sqrt{7}
\][/tex]
Factor out the common coefficient [tex]\( 5 \)[/tex]:
[tex]\[
= 5 (\sqrt{5} - \sqrt{7})
\][/tex]
The expression simplifies to:
[tex]\[
5 (\sqrt{5} - \sqrt{7}) \approx -2.0484166678240046
\][/tex]
Thus, we have the simplified forms:
1. [tex]\( (\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) \approx -1.2889720753539853 \)[/tex]
2. [tex]\( \sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21} \approx -1.288972075353985 \)[/tex]
3. [tex]\( 2 \sqrt{5} - 2 \sqrt{7} \approx -0.8193666671296018 \)[/tex]
4. [tex]\( 2 \sqrt{5} + 3 \sqrt{5} - 2 \sqrt{7} - 3 \sqrt{7} \approx -2.0484166678240046 \)[/tex]
