Nth root

An n-th 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:

${\displaystyle k^{n}=r}$

This is the graph for ${\displaystyle y={\sqrt {x}}}$. It is a square root.

This is ${\displaystyle y={\sqrt[{3}]{x}}}$. It is a cube root.

(for the meaning of ${\displaystyle k^{n}}$, see Exponentiation.)

We write the nth root of r as ${\displaystyle {\sqrt[{n}]{r}}}$.^[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, ${\displaystyle {\sqrt[{3}]{8}}=2}$ because ${\displaystyle 2^{3}=8}$. The 8 in that example is called the radicand, the 3 is called the index, and the check-shaped part is called the radical symbol or radical sign.

Roots and powers can be changed as shown in ${\displaystyle {\sqrt[{b}]{x^{a}}}=x^{\frac {a}{b}}=({\sqrt[{b}]{x}})^{a}=(x^{a})^{\frac {1}{b}}}$.

The product property of a radical expression is the statement that ${\displaystyle {\sqrt {ab}}={\sqrt {a}}\times {\sqrt {b}}}$. The quotient property of a radical expression is the statement ${\displaystyle {\sqrt {\tfrac {a}{b}}}={\tfrac {\sqrt {a}}{\sqrt {b}}}}$.^[3], b != 0.

Simplifying

This is an example of how to simplify a radical.

${\displaystyle {\sqrt {8}}={\sqrt {4\times 2}}={\sqrt {4}}\times {\sqrt {2}}=2{\sqrt {2}}}$

If two radicals are the same, they can be combined. This is when both of the indexes and radicands are the same.^[4]

${\displaystyle 2{\sqrt {2}}+1{\sqrt {2}}=3{\sqrt {2}}}$

${\displaystyle 2{\sqrt[{3}]{7}}-6{\sqrt[{3}]{7}}=-4{\sqrt[{3}]{7}}}$

This is how to find the perfect square and rationalize the denominator.

${\displaystyle {\frac {8x}{{\sqrt {x}}^{3}}}={\frac {8{\cancel {x}}}{{\cancel {x}}{\sqrt {x}}}}={\frac {8}{\sqrt {x}}}={\frac {8}{\sqrt {x}}}\times {\frac {\sqrt {x}}{\sqrt {x}}}={\frac {8{\sqrt {x}}}{{\sqrt {x}}^{2}}}={\frac {8{\sqrt {x}}}{x}}}$

Related pages

• Geometric mean
• Rationalization (mathematics)
• Square number

References

1. "List of Arithmetic and Common Math Symbols". Math Vault. 2020-03-17. Retrieved 2020-09-22.
2. Weisstein, Eric W. "nth Root". mathworld.wolfram.com. Retrieved 2020-09-22.
3. "nth Roots". www.mathsisfun.com. Retrieved 2020-09-22.
4. "Add and Subtract Radicals". mathbitsnotebook.com. Retrieved March 14, 2018.