Answer :
The simplified product is 18x -6[tex]x^4[/tex] + 24x²√2 - [tex]8x^5[/tex]√2.
or 18x -[tex]x^4[/tex] √30x + 24x²√2 - [tex]8x^5[/tex]√2
What is Radical?
The √ symbol that is used to denote square root or nth roots. Radical Expression - A radical expression is an expression containing a square root. Radicand - A number or expression inside the radical symbol.
We have
(√6x² + 4√8x³)(√9x - x√5[tex]x^5[/tex])
Now, simplifying the radical is
√6x² = √6 x
4√8x³ = 4(2x) √2x= 8x√2x
√9x = 3√x
x√5[tex]x^5[/tex]= x³ √x
Now, the simplified expression is
(√6x² + 4√8x³)(√9x - x√5[tex]x^5[/tex])
= (√6 x + 8x√2x) (3√x - x³ √x)
= 6(3)(x) -6[tex]x^4[/tex] + 24x²√2 - [tex]8x^5[/tex]√2
= 18x -6[tex]x^4[/tex] + 24x²√2 - [tex]8x^5[/tex]√2
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The correct answer is: D) [tex]$3 x \sqrt{6 x}-x^4 \sqrt{30 x}+24 x^2 \sqrt{2}-8 x^5 \sqrt{10}$[/tex].
To simplify the given product, let's first expand it:
[tex]\left(\sqrt{6 x^2}+4 \sqrt{8 x^3}\right)\left(\sqrt{9 x}-x \sqrt{5 x^5}\right)[/tex]
[tex]=\sqrt{6 x^2} \cdot \sqrt{9 x}+\sqrt{6 x^2} \cdot\left(-x \sqrt{5 x^5}\right)+4 \sqrt{8 x^3} \cdot \sqrt{9 x}-4 \sqrt{8 x^3} \cdot\left(x \sqrt{5 x^5}\right)[/tex]
[tex]=3 x \sqrt{6 x}-x^4 \sqrt{30 x}+24 x^2 \sqrt{2}-8 x^5 \sqrt{10}[/tex]
Thus, the fourth option matches the simplified product.
Complete Question:
What is the following simplified product? Assume [tex]$x \geq 0$[/tex].
A) [tex]$\left(\sqrt{6 x^2}+4 \sqrt{8 x^3}\right)\left(\sqrt{9 x}-x \sqrt{5 x^5}\right)$[/tex]
B) [tex]$3 x \sqrt{6 x}+x^4 \sqrt{30 x}+24 x^2 \sqrt{2 x}+8 x^5 \sqrt{10 x}$[/tex]
C) [tex]$3 x \sqrt{6 x}+x^4 \sqrt{30 x}+24 x^2 \sqrt{2}+8 x^5 \sqrt{10}$[/tex]
D) [tex]$3 x \sqrt{6 x}-x^4 \sqrt{30 x}+24 x^2 \sqrt{2}-8 x^5 \sqrt{10}$[/tex]
E) [tex]$3 x \sqrt{6 x}-x^4 \sqrt{30 x}+24 x^2 \sqrt{2 x}-8 x^5 \sqrt{10 x}$[/tex]
