Answer :
We start with the expression
[tex]$$
\frac{8x^6 \sqrt{200x^{13}}}{2x^5 \sqrt{32x^7}}.
$$[/tex]
Below is a step-by-step simplification.
1. First, rewrite the radicals by factoring out perfect squares.
• For the numerator, notice that
[tex]$$
200x^{13} = 4\cdot25\cdot2\,(x^6)^2\,x.
$$[/tex]
Thus,
[tex]$$
\sqrt{200x^{13}} = \sqrt{4\cdot25\cdot2\,(x^6)^2\,x} = 2\cdot5\,x^6\sqrt{2x} = 10x^6\sqrt{2x}.
$$[/tex]
• For the denominator, observe that
[tex]$$
32x^7 = 16\cdot2\,(x^3)^2\,x.
$$[/tex]
So,
[tex]$$
\sqrt{32x^7} = \sqrt{16\cdot2\,(x^3)^2\,x} = 4\,x^3\sqrt{2x}.
$$[/tex]
2. Substitute these into the original expression:
• The numerator becomes
[tex]$$
8x^6\cdot\left(10x^6\sqrt{2x}\right)=80x^{12}\sqrt{2x},
$$[/tex]
• The denominator becomes
[tex]$$
2x^5\cdot\left(4x^3\sqrt{2x}\right)=8x^{8}\sqrt{2x}.
$$[/tex]
3. Divide the numerator by the denominator:
[tex]$$
\frac{80x^{12}\sqrt{2x}}{8x^8\sqrt{2x}} = \frac{80}{8}\cdot x^{12-8}\cdot \frac{\sqrt{2x}}{\sqrt{2x}}.
$$[/tex]
Here, [tex]$\sqrt{2x}$[/tex] cancels out, leaving
[tex]$$
10x^4.
$$[/tex]
Thus, the final simplified expression is
[tex]$$
10x^4.
$$[/tex]
[tex]$$
\frac{8x^6 \sqrt{200x^{13}}}{2x^5 \sqrt{32x^7}}.
$$[/tex]
Below is a step-by-step simplification.
1. First, rewrite the radicals by factoring out perfect squares.
• For the numerator, notice that
[tex]$$
200x^{13} = 4\cdot25\cdot2\,(x^6)^2\,x.
$$[/tex]
Thus,
[tex]$$
\sqrt{200x^{13}} = \sqrt{4\cdot25\cdot2\,(x^6)^2\,x} = 2\cdot5\,x^6\sqrt{2x} = 10x^6\sqrt{2x}.
$$[/tex]
• For the denominator, observe that
[tex]$$
32x^7 = 16\cdot2\,(x^3)^2\,x.
$$[/tex]
So,
[tex]$$
\sqrt{32x^7} = \sqrt{16\cdot2\,(x^3)^2\,x} = 4\,x^3\sqrt{2x}.
$$[/tex]
2. Substitute these into the original expression:
• The numerator becomes
[tex]$$
8x^6\cdot\left(10x^6\sqrt{2x}\right)=80x^{12}\sqrt{2x},
$$[/tex]
• The denominator becomes
[tex]$$
2x^5\cdot\left(4x^3\sqrt{2x}\right)=8x^{8}\sqrt{2x}.
$$[/tex]
3. Divide the numerator by the denominator:
[tex]$$
\frac{80x^{12}\sqrt{2x}}{8x^8\sqrt{2x}} = \frac{80}{8}\cdot x^{12-8}\cdot \frac{\sqrt{2x}}{\sqrt{2x}}.
$$[/tex]
Here, [tex]$\sqrt{2x}$[/tex] cancels out, leaving
[tex]$$
10x^4.
$$[/tex]
Thus, the final simplified expression is
[tex]$$
10x^4.
$$[/tex]