Answer:
[tex]\sqrt[n]{a} =a^{\frac{1}{n}}[/tex]
Explanation:
Roots of real numbers can be represented by radicals or by exponents.
First, I present some examples to show how exponents and radicals are related, and then generalize.
[tex]\sqrt{4}=4^{(\frac{1}{2})}=(2^2)^\frac{1}{2}=(2)^{\frac{2}{2}}=2^1=2\\\\\\\sqrt{25}=(25)^{\frac{1}{2}}=(5^2)^{\frac{1}{2}}=(5)^{\frac{2}{2}}=5^1=5[/tex]
[tex]\sqrt[3]{8}=(8)^{\frac{1}{3}}=(2^3)^{\frac{1}{3}}=(2)^{\frac{3}{3}}=2^1=2[/tex]
When you write 5² = 25, then 5 is the square root of 25.
And in general, if n is a positive integer and [tex]a^n=x[/tex] , then [tex]a[/tex] is the nth root of x.
Also, if n even (and positive) and [tex]a[/tex] is positive, then [tex]a^{\frac{1}{n}}[/tex] is the positive nth root of [tex]a[/tex]
Thus,
[tex]\sqrt[n]{a} =a^{\frac{1}{n}}[/tex]
