Is there a good way to get rid of the dry skin on my arms?

Cream. If it is just common dry skin try thick cream instead of watery lotion but if persists see your doctor to make sure it is not something more like eczema or psoriasis.
Avoid sun damage. The sun causes skin to get dry and leathery. Drink plenty of water and apply body moisturizers twice daily. After taking a shower apply body oil or lotion to lock in the moisture. Do not sit in the hot tub, swimming pool, bath tub or any other body of water for long periods of time.

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Is there a good way to get rid of the dry skin on my face?

Hydration. Skin that is dry may be related to allergies, or the use of dying cleansers or dry weather. Hydration of the skin and the use of mild cleansers and tepid water is essential/ remember to put moisturizer on slightly moist skin. Read more...