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].
