Roots and Radicals

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separating dimensions

While polynomials and rational functions only include terms with positive integer powers, power functions, by themselves, include any real number power. The noninteger powers belong to a category called roots and radicals.

Roots

$x^{\tfrac {1}{n}}$ is called the nth root of x, where $n$ is a natural number.

odd roots

When $n$ is odd, $x^{\tfrac {1}{n}}$ is defined to be the unique real number, $r$ , such that $r^n = x$ . The domain is all real numbers.

even roots

When $n$ is even, $x^{\tfrac {1}{n}}$ is defined to be the unique nonnegative real number, $r$ , such that $r^n = x$ . The domain is all nonnegative real numbers.

Radicals

An alternate point-of-view uses radical notation rather than exponents.

$\sqrt [n]{x} = x^{\tfrac {1}{n}}$

where $\sqrt { }$ is called the radical symbol.

$\sqrt [n]{x}$ is called the nth root of x.

Algebra Rules

The nth-root is an exponent, so it follows all of the exponent rules.

$a^n \cdot a^m = a^{n+m}$
$\frac {a^n}{a^m} = a^{n-m}$
$(a^n)^m = a^{n \cdot m}$
$a^n \cdot b^n = (a \cdot b)^n$
$\frac {a^n}{b^n} = \left (\frac {a}{b}\right )^n$

Similar rules for radicals.

$\sqrt [n]{a} \cdot \sqrt [n]{b} = \sqrt [n]{a \cdot b}$
$\frac {\sqrt [n]{a}}{\sqrt [n]{b}} = \sqrt [n]{\frac {a}{b}}$
$\sqrt [m]{\sqrt [n]{a}} = \sqrt [nm]{a}$

Graphs

Negative numbers raised to odd powers result in negative numbers, which means every real number has an odd root. The domain of odd roots or radicals is all real numbers.

The graph of $y = R(t) = t^{\tfrac {1}{3}} = \sqrt [3]{t}$

Negative numbers raised to even powers result in positive numbers, just like positive numbers raised to even power. This makes going backwards a problem.

Should $\sqrt {4} = -2$ or $\sqrt {4} = 2$ ? When squared, both $-2$ and $2$ give $4$ .

We are forced to choose between two even roots. We pick positive numbers . The domain and range of even roots are both $[0, \infty )$ .

The graph of $y = R(t) = t^{\tfrac {1}{2}} = \sqrt {t}$

There are no horizontal asymptotes on these graphs. The graph keeps moving up or down unbounded. The root and radical functions are unbounded. Their values tend to $\pm \infty$ as the domain moves out towards $\pm \infty$ .

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more examples can be found by following this link
More Examples of Elementary Functions