### Video Transcript

Find ๐ฆ given the geometric mean between two and ๐ฆ is 10.

First letโs recall the definition of the geometric mean. The geometric mean of two positive numbers ๐ and ๐ is the square root of their product, ๐๐. In this question, weโve been told that the geometric mean is 10. Weโve also been told one of the numbers is two. And we want to work backwards in order to find the other.

So using the information in the question and our definition of the geometric mean, we can write down that the square root of the product of the two numbers two ๐ฆ is equal to 10. This gives us an equation that we can solve in order to find the value of ๐ฆ.

As there is a square root on the left-hand side of the equation, the first step is to square both sides. This cancels out the square root and gives two ๐ฆ is equal to 10 squared, which is 100. Finally, to find the value of ๐ฆ, we need to divide both sides of this equation by two. This gives the solution to the problem: ๐ฆ is equal to 50.