Question

How are exponents and radicals used to present roots of real numbers

Answer 1

Answer:

       $\sqrt[n]{a} =a^{\frac{1}{n}}$

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.

      $\sqrt{4}=4^{(\frac{1}{2})}=(2^2)^\frac{1}{2}=(2)^{\frac{2}{2}}=2^1=2\\\\\\\sqrt{25}=(25)^{\frac{1}{2}}=(5^2)^{\frac{1}{2}}=(5)^{\frac{2}{2}}=5^1=5$

       $\sqrt[3]{8}=(8)^{\frac{1}{3}}=(2^3)^{\frac{1}{3}}=(2)^{\frac{3}{3}}=2^1=2$

When you write 5² = 25, then 5 is the square root of 25.

And in general, if n is a positive integer and $a^n=x$ , then $a$ is the nth root of x.

Also, if n even (and positive)  and $a$ is positive, then $a^{\frac{1}{n}}$   is the positive nth root of $a$

Thus,

       $\sqrt[n]{a} =a^{\frac{1}{n}}$