In the triangle ABC, AB = 7 dm, BC = 10 dm, angle B = 45 degrees. Find AC.

You know 2 sides and the angle opposite to the unknown side. You can apply the cosine theorem (The square of the side of a triangle is the sum of the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them).
AC ^ 2 = AB ^ 2 + BC ^ 2-2 * AB * BC * cosB, cos45 = (√2) / 2
AC ^ 2 = 7 ^ 2 + 10 ^ 2 – 2 * 7 * 10 * (√ 2) / 2
AC ^ 2 = 49 + 100-70√2
AC ^ 2 = 149-70√2
AC = √149-70√2
Answer: AC = √ (149-70√2)