The exponent of a number says how many times to use the number in a multiplication. 3^3 would be 27, and 3^-3 would be 1/3^3 which is 1/27. An exponent of 1/2 is actually square root, an exponent of 1/3 is cube root, an exponent of 1/4 is 4th root, and so on. This is because (9^1/2)^2 is 9^(1/2+1/2), which is 9^1, which is 9. That means 9^1/2 is the square root of 9, or 3. A fractional exponent like 1/n means to take the n-th root.
But you might say, "What about 27^4/3?" The answer would be 81. I will prove this in 2 ways.
1: 27^4/3 = the fourth root of 27^3, which is the fourth root of 531441, which is 81.
2: 27^4/3 = 27^(1/3*4) = (the cube root of 27)^4 = 3^4 = 81
You can see that the second way is WAY better!
So these rules can work:
x^n = x*x*x*x*...*x (n times)
x^-n = 1/x^n
x^1/n = the nth root of x
x^m/n = x^(1/n*m) = (the nth root of x)^m
I hope you enjoyed this episode of MatheMagics by Siyoung. See you in the next episode!
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