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Exponents
Where any value base is being "raised to the power" , the value of is said to be
the exponent. It is the number of times base is to be multiplied by itself; that is:
\(a^y = \displaystyle\prod_{k=1}^{y} = a\cdot a\cdot\ldots\cdot a\)
Any nonzero value with a zero exponent is equal to :
\(a^0 = 1\) if \(a\) is positive, \(-1\) if \(a\) is negative, or undetermined if \(a = 0\).
Values with negative exponents are reciprocals of the base raised to the power of the associated positive exponent:
\(a^{-1}=\frac{1}{a^1}\), \(a^{-2}=\frac{1}{a^2}\), etc.
Values with fractional exponents are nth-root functions in the form of \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\), therefore
\(x^{\frac{1}{n}} = \sqrt[n]{x}\).
Nth-Roots
An nth-root function is the inverse of a power function; that is, if \(x^n=y\), then \(\sqrt[n]{y}=x\). For example, if
we square the number \(5\), (that is, we perform the operation \(x=5^2\)), we get a result of \(5\cdot 5=25\). Therefore,
the square root of \(25\) is \(5\). Mathematically, we would write this as (\(\sqrt{25}=5\)). Likewise, if we cube the
number \(5\), we get \(125\), so (\(\sqrt[3]{125}=5\)).
Exponential Identities
This list of identities shows equivalent forms of many power functions and will help when working problems involving exponents and roots.
\(x^m = \frac{1}{x^{-m}}\)
\(x^{-m} = \frac{1}{x^m}\)
\(x^{\frac{1}{n}} = \sqrt[n]{x}\)
\(x^{-\frac{1}{n}} = \frac{1}{\sqrt[n]{x}} = \sqrt[n]{\frac{1}{x}}\)
\(x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\)
\(x^{-\frac{m}{n}} = \frac{1}{\sqrt[n]{x^m}} = \sqrt[n]{\frac{1}{x^m}} = \sqrt[n]{x^{-m}}\)
\(x^{m+n} = x^m\cdot x^n\)
\(x^{m+\frac{1}{n}} = x^m\cdot \sqrt[n]{x}\)
\(x^{m-n} = \frac{x^m}{x^n}\)
\(x^{mn} = (x^m)^n\)
\(x^ny^n = (xy)^n\)
\(\frac{x^n}{y^n} = (\frac{x}{y})^n\)
\(\sqrt[n]{pq} = \sqrt[n]{p}\cdot \sqrt[n]{q}\)
\(\sqrt[n]{\frac{p}{q}} =\frac{\sqrt[n]{p}}{\sqrt[n]{q}}\)
\(\sqrt[n]{p^nq} = p\sqrt[n]{q}\)
\(\frac{\sqrt[n]{x}}{x} = \frac{1}{\sqrt[n]{x}}\)
\(x^n = a^{n\log_a{x}}\)
\(\sqrt[n]{x} = a^{\frac{\log_a{x}}{n}}\)
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