The base and exponent form a pair. You can chose any positive base and that will determine the exponent.243 = 49^1.5 that is base = 49, exponent = 1.5)or
243 = 10^2.5353 (approx)
or
243 = 7^3.
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The base of 243 is 3, as 3 raised to the power of 5 equals 243. The exponent of 243 is 5, representing the number of times the base (3) is multiplied by itself to equal 243. In exponential form, 243 can be written as 3^5.
The two are related. The answer could be base 2, exponent 18 or base 8, exponent 6 or base 10, exponent 5.4185 or base 262144, exponent 1 or base 68,719,476,736 and exponent 0.5
The base could be 11 and the exponent 2, giving 112 But, it could equally be base = 14641, and exponent = 0.5, or base = 10, and exponent = 2.082785 (approx)
The base is 7 and the exponent is 3.
You can define any base you like and calculate an appropriate exponent or, you can pick an exponent and calculate the base. So you can have base 25, with exponent 2 or base 5 and exonent 4 or base e (the base for natural logarithms) and exponent 6.437752 (to 6 dp) or base 10 and exponent 2.795880 (to 6 dp) or base 2 and exponent 9.287712 etc or base 8.54988 (to 3 dp) and exponent 3 or base 3.623898 (to 3 dp) and exponent 5 etc There is no need for the base to be an integer or even rational. Probably the most important bases in advanced mathematics is e, which is a transcendental number. Similarly, there is no need for the exponent to be an integer.
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