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

Sagot :

To find the product of the expressions [tex]\((-3 + \sqrt{5})\)[/tex] and [tex]\((7 + \sqrt{3})\)[/tex], we will follow these steps:

1. Express the product:
[tex]\[ (-3 + \sqrt{5})(7 + \sqrt{3}) \][/tex]

2. Use the distributive property (FOIL method): Multiply each term in the first expression by each term in the second expression.

3. Calculate each part:

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

Breaking it down:

- [tex]\( (-3) \cdot 7 = -21 \)[/tex]
- [tex]\( (-3) \cdot \sqrt{3} = -3\sqrt{3} \)[/tex]
- [tex]\( \sqrt{5} \cdot 7 = 7\sqrt{5} \)[/tex]
- [tex]\( \sqrt{5} \cdot \sqrt{3} = \sqrt{15} \)[/tex]

4. Combine all terms:

[tex]\[ -21 - 3\sqrt{3} + 7\sqrt{5} + \sqrt{15} \][/tex]

However, when we combine these terms using the precise arithmetic operations, the final numerical result is approximately:

[tex]\[ -6.6706932340006855 \][/tex]

Thus, the product of [tex]\((-3 + \sqrt{5})\)[/tex] and [tex]\((7 + \sqrt{3})\)[/tex] is approximately [tex]\(-6.6706932340006855\)[/tex].