Find the geometric mean of three squared and six squared.
We’ll begin by recalling the definition of the geometric mean. For a pair of numbers 𝑎 and 𝑏 with the same sign, the geometric mean of 𝑎 and 𝑏 is the square root of 𝑎𝑏. We can only find the geometric mean of two numbers with the same sign because if the two numbers were of opposite signs, then the product would be negative. And the square root of a negative number gives a nonreal result.
We’re looking to find the geometric mean of the two numbers three squared and six squared. Now three squared is nine and six squared is 36. So, these two numbers are both positive. They, therefore, have the same sign, and so we can find their geometric mean. Now there are two ways that we could approach this problem. One method is to evaluate three squared and six squared. The geometric mean of three squared and six squared is the square root of their product, so we have the square root of nine multiplied by 36. Nine multiplied by 36 is 324. And we could use a calculator. Or perhaps we know that the square root of 324 is 18.
The other method we could use, which is perhaps the smarter of the two methods here, is to recall one of our rules of surds or radicals. The square root of a product 𝑎𝑏 is equal to the square root of 𝑎 multiplied by the square root of 𝑏. So the square root of three squared multiplied by six squared is just the square root of three squared multiplied by the square root of six squared. Three and six are both positive numbers. And as squaring and square rooting are inverse operations for positive numbers, the square root of three squared is three and the square root of six squared is six. So we have three multiplied by six, which once again is equal to 18. The geometric mean then of three squared and six squared is 18.