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nth root
An nth root of a number r is a number which, if n copies are multiplied together, makes r. It is also called a radical or a radical expression. It is a number k for which the following equation is true:
 [math]k^n=r[/math]
(for the meaning of [math]k^n[/math], see Exponentiation.)
We write the nth root of r as [math]\sqrt[n]{r}[/math].^{[1]} If n is 2, then the radical expression is a square root. If it is 3, it is a cube root.^{[2]}^{[3]} Other values of n are referred to using ordinal numbers, such as fourth root and tenth root.
For example, [math]\sqrt[3]{8} = 2[/math] because [math]2^3 = 8[/math]. The 8 in that example is called the radicand, the 3 is called the index, and the checkshaped part is called the radical symbol or radical sign.
Roots and powers can be changed as shown in [math]\sqrt[b]{x^a} = x^\frac{a}{b} = (\sqrt[b]{x})^a = (x^a)^\frac{1}{b}[/math].
The product property of a radical expression is the statement that [math]\sqrt{ab} = \sqrt{a} \times \sqrt{b}[/math]. The quotient property of a radical expression is the statement [math]\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}[/math].^{[3]}
Simplifying
This is an example of how to simplify a radical.
 [math]\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}[/math]
If two radicals are the same, they can be combined. This is when both of the indexes and radicands are the same.^{[4]}
 [math]2\sqrt{2} + 1\sqrt{2} = 3\sqrt{2}[/math]
 [math]2\sqrt[3]{7}  6\sqrt[3]{7} = 4\sqrt[3]{7}[/math]
This is how to find the perfect square and rationalize the denominator.
 [math]\frac{8x}{\sqrt{x}^3} = \frac{8\cancel{x}}{\cancel{x}\sqrt{x}} = \frac{8}{\sqrt{x}} = \frac{8}{\sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{8\sqrt{x}}{\sqrt{x}^2} = \frac{8\sqrt{x}}{x}[/math]
Related pages
References
 ↑ "List of Arithmetic and Common Math Symbols" (in enUS). 20200317. https://mathvault.ca/hub/highermath/mathsymbols/commonmathsymbols/.
 ↑ Weisstein, Eric W.. "nth Root" (in en). https://mathworld.wolfram.com/nthRoot.html.
 ↑ ^{3.0} ^{3.1} "nth Roots". https://www.mathsisfun.com/numbers/nthroot.html.
 ↑ "Add and Subtract Radicals". https://mathbitsnotebook.com/Algebra1/Radicals/RADAddSubtract.html.

