## Will the power of x increase or decrease its absolute value (i.e. the value regardless of the sign, + or -)?

Well, there are two exceptions to this question. X cannot be 0 or 1 because 0*0=0, and 1*1=1. No matter how many times you multiply 0 by itself, you will always get 0, and no matter how many times you multiply 1 by itself, you will always get 1. That's why the power of x will never change its value if x is 0 or 1. Now that we realize the two exceptions of 0 and 1 for x, x would have to be in one of two certain ranges: 0<x<1 or x>1.

If 0<x<1, then that would mean that x is a proper fraction when the numerator is smaller than the denominator (e.g. 5/6). Let's use the easiest fraction value for x, 1/2, and the easiest power of x, x^2. Plug in the value of x, and you will get x^2=(1/2)^2. This will multiply the fraction of 1/2 twice by itself: (1/2)*(1/2). Now, since any number times 1 is that number, (1/2)*1=1/2 so that 1/2 remains the same. So if the second term is less than 1, it will make the first term smaller than itself as (1/2)*(1/2)=1/4. Therefore, the power of x will decrease its value if 0<x<1.

If x>1, then x would have to be anything beyond the whole number of 1 in any of the following forms: improper fraction (e.g. 3/2 because the numerator is greater than the denominator), mixed number (e.g. 1 1/2), decimal (e.g. 1.01 with at least one non-zero digit on each side of the decimal point), or whole number (e.g. 2). Let's try using for a value of x a decimal number of 1.1 as one of those forms. Plug in the value of x, and you will get x^2=(1.1)^2. This will multiply the fraction of 1.1 twice by itself: (1.1)*(1.1). Now, since any number times 1 is that number, (1.1)*1=1.1 so that 1.1 remains the same. So if the second term is greater than 1, it will make the first term larger than itself as (1.1)*(1.1)=1.21. Therefore, the power of x will increase its value if x>1.

In conclusion, the power of x will either increase or decrease its value as it depends on two specific ranges: 0<x<1 or x>1. If the value of x is in the former range, then the exponent will decrease its value, and the greater the exponent, the closer to 0 x will get. Otherwise, if the value of x is in the latter range, then the exponent will increase its value, and the higher the exponent, the higher the value of x. Otherwise, the value of x will always be 0 or 1 whichever one of these values it is. Consequently, x can be any value but 0 and 1 for its power to alter its value.