9514 1404 393
Answer:
(x^(1/2))(x^(1/2)) = x^(1/2 +1/2) = x^1 = x
Step-by-step explanation:
The rule of exponents is ...
(x^a)(x^b) = x^(a+b)
From which ...
(x^a)(x^a) = x^(a+a) = x^(2a)
So, if we want two identical factors that have a product of x = x^1, then the exponents of those factors will be such that ...
x^(2a) = x^1
2a = 1
a = 1/2
The square root is defined as one of two identical factors that have a product equal to the specified value. That is ...
(√x)(√x) = x
Above, we have shown that ...
(x^(1/2))(x^(1/2)) = x
so, we can conclude ...
√x = x^(1/2)
_____
Additional comment
In like fashion, we can show that the n-th root of a number is the same as that number to the 1/n power. It's really a matter of definition. Since the square of x^(1/2) is x, we call x^(1/2) the square root. It is used commonly enough that it has its own symbol: √x.
