The gemstone shown is a square pyramid that has a base with sides 3.4 inches long. the slant height of the pyramid is 3.8 inches. find the surface area of the gemstone?

To find surface area we add up the areas of each face of the pyramid.  The area of the square base is given by (3.4)(3.4) = 11.56 in².

To find the area of each triangular face, we first need to find the height of each triangle.  We have the slant height and the base.  The slant height serves as one of the sides of the triangle.  We can draw the height from the top vertex to the base and use the Pythagorean Theorem to find the height.  This height we drew splits the base in half, so in our Pythagorean Theorem we have:
(1.7)² + b² = (3.8)²
(The half of the base will be one leg and the slant height is the hypotenuse)
This gives us 2.89 + b² = 14.44
Subtract 2.89 from each side to cancel it:
2.89 + b² - 2.89 = 14.44 - 2.89
b² = 11.55
Take the square root of both sides and we have 
b = 3.4
The formula for the area of a triangle is A = (1/2)bh.  The base of our triangle is 3.4 and the height we just found is 3.4.
(1/2)(3.4)(3.4) = 5.78
The surface area will be 11.56 + 3.4 + 3.4 + 3.4 + 3.4 (there are 4 triangular faces) or 34.68 in²