6 A solid steel cube of side 8 cm is melted down and recast into six identical spheres. Find the radius of each of the six spheres.

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Let the side of the cube be $$ s = 8 $$ cm. The volume of the cube is  
$$
V_{\text{cube}} = s^3 = 8^3 = 512 \text{ cm}^3.
$$

When the cube is melted and recast into six identical spheres, the volume of each sphere is  
$$
V_{\text{sphere}} = \frac{512}{6} = \frac{256}{3} \text{ cm}^3.
$$

The volume of a sphere in terms of its radius $$ r $$ is given by  
$$
V_{\text{sphere}} = \frac{4}{3}\pi r^3.
$$

Setting the two expressions for the volume equal gives  
$$
\frac{4}{3}\pi r^3 = \frac{256}{3}.
$$

Multiply both sides by 3 to simplify:  
$$
4\pi r^3 = 256.
$$

Solve for $$ r^3 $$ by dividing both sides by $$ 4\pi $$:  
$$
r^3 = \frac{256}{4\pi} = \frac{64}{\pi}.
$$

Taking the cube root on both sides, we obtain  
$$
r = \left(\frac{64}{\pi}\right)^{1/3}.
$$

Since $$ 64 = 4^3 $$, we can simplify further:  
$$
r = \frac{4}{\pi^{1/3}}.
$$

Thus, the radius of each sphere is  
$$
\boxed{\frac{4}{\pi^{1/3}} \text{ cm}}.
$$
The radius of each sphere is $$ \frac{4}{\pi^{1/3}} $$ cm.

6 A solid steel cube of side 8 cm is melted down and recast into six identical spheres. Find the radius of each of the six spheres.

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