Simplify the square root of three to the 11th power to the eighth power over the square root of three to the eighth power to the 11th power.
The first thing that I wanna do is take the square root of three and rewrite it as three to the one-half power. That means in the numerator, I’ll have three to the one-half power to the 11th power to the eighth power and in the denominator three to the one-half power to the eighth power to the 11th power.
And then we want to consider what it is to take a power of a power. When we have 𝑥 to the 𝑎 to the 𝑏 power, that’s the same thing as saying 𝑥 to the 𝑎 times 𝑏. And that means we can say the numerator is three to the one-half times 11 times eight power. And then the denominator would be three to the one-half times eight times 11.
Are you noticing something yet? The numerator and the denominator have the same exponents. And they’re just being multiplied in a different order. One-half times 11 times eight equals 44. And one-half times eight times 11 also equals 44. And three to the 44th power over three to the 44th power equals one. Anything divided by itself is one.