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Find the geometric mean of (𝑎 − 19) and (𝑎 + 19).

Find the geometric mean of 𝑎 minus 19 and 𝑎 plus 19.

The geometric mean of two values 𝑎 and 𝑏 is the square root of 𝑎 times 𝑏, which means we need to multiply 𝑎 minus 19 times 𝑎 plus 19 and then take the square root of that value.

Our first step will start by multiplying. We need to FOIL these two expressions. 𝑎 times 𝑎 equals 𝑎 squared. 𝑎 times 19 equals 19𝑎. Negative 19 times 𝑎 equals negative 19𝑎. And negative 19 times 19 equals negative 361.

This middle term says 19𝑎 minus 19𝑎 and that equals zero. They cancel each other out. We now have 𝑎 squared minus 361. And we must take the square root of that value.

And our geometric mean is then the square root of 𝑎 squared minus 361.

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