Answer:
[tex]\large\boxed{(3+\sqrt{-16})(6\sqrt{-64})=-192+144i}[/tex]
Step-by-step explanation:
[tex]\sqrt{-1}=i\to i^2=-1\\\\(3+\sqrt{-16})(6\sqrt{-64})=(3+\sqrt{(16)(-1)})(6\sqrt{(64)(-1)})\\\\=(3+\sqrt{16}\cdot\sqrt{-1})(6\cdot\sqrt{64}\cdot\sqrt{-1})=(3+4i)\bigg((6)(8i)\bigg)\\\\=(3+4i)(48i)\qquad\text{use the distributive property}\ (b+c)a=ba+ca\\\\=(3)(48i)+(4i)(48i)=144+192i^2\\\\=144i+192(-1)=-192+144i[/tex]
The value of the given expression (3+√-16)(6√-64) is (-192+144i).
As the given two factors are complex numbers, therefore, we must know about the value of i
i = √(-1)
i² = (-1)
The solution of the product,
[tex](3+\sqrt{-16})(6\sqrt{-64})\\\\ = (3\times 6\sqrt{-64}) + (\sqrt{-16}\times 6\sqrt{-64})\\\\= (3+4i)(6\cdot 8i)\\\\= (3+4i)(48i)\\\\= 144i + 192(i)^2\\\\= 144i + 192(\sqrt{-1})^2\\\\= 144i + 192(-1)\\\\= -192+144i[/tex]
Hence, the value of the given expression (3+√-16)(6√-64) is (-192+144i).
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