Simplify the following expression:

[tex]\[ (\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) \][/tex]

A. [tex]\(\sqrt{10}+\sqrt{15}-\sqrt{14}-\sqrt{21}\)[/tex]

B. [tex]\(2\sqrt{5}-2\sqrt{7}\)[/tex]

C. [tex]\(2\sqrt{5}+3\sqrt{5}-2\sqrt{7}-3\sqrt{7}\)[/tex]

D. [tex]\(\sqrt{5}+\sqrt{-2}\)[/tex]



Answer :

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]