Simplify the square of 100𝑥 to the sixteenth.
So looking at the square root of 100𝑥 to the sixteenth, most of us know the square root of 100. The square root of 100 is equal to 10 because 10 times 10 equals 100, and 10 times 10 is 10 squared. So if you would take the square root of 10 squared or 100, the square root goes away, and that’s just equal to 10. So 100 is 10 times 10, so we know we have 10 in our answer.
Now 𝑥 to the sixteenth, that’s 𝑥 to the something times 𝑥 to the something, and they have to match, like 10 times 10. So we need 𝑥 to a number and 𝑥 to the same number.
Now when you multiply bases with exponents, you add your exponents. So what two numbers that are the same add to be sixteen? Or we can just take 16 and split it evenly, 16 divided by two. That would be 𝑥 to the eighth times 𝑥 to the eighth, because 𝑥 to the eighth times 𝑥 to the eighth is the same thing as 𝑥 to the eight plus eight, which is 𝑥 to the sixteenth.
Therefore, the square root of 100𝑥 to the sixteenth would be 10𝑥 to the eighth. We can always double-check our work by using the inverse function. So instead of a square root, we square it and we make sure that we would get 100𝑥 to the sixteenth.
Well 10 squared is 100, and we use the multiplication rule for exponents, so we take 𝑥 to the eighth or we take eight times the two, so we do indeed get 100𝑥 to the sixteenth.