Rational Exponents are exponents where the numerator is a power and the denominator is a root. They can be expressed as:
NOTE: In a rational exponent, \(n ≠ 0\).
Rational Exponents can be written in radical notation and vice versa:
For example, the rational exponent \(\textcolor{red}{4}^{\textcolor{green}{1}/\textcolor{blue}{2}}\) can be rewritten as \(\sqrt{\textcolor{red}{4}}\). Likewise, the radical \(\sqrt[\textcolor{blue}{3}]{\textcolor{red}{8}^{\textcolor{green}{2}}}\) can be written \(\textcolor{red}{8}^{\textcolor{green}{2}/\textcolor{blue}{3}}\).
Rewrite the following in a different notation. Simplify if possible:
i. We can rewrite this as in radical notation by placing the numerator (\(2\)) as the power and the denominator (\(7\)) as the root. Then, we can simplify:
\(= \sqrt[\textcolor{blue}{7}]{\textcolor{red}{6}^{\textcolor{green}{2}}}\)
\(= \sqrt[\textcolor{blue}{7}]{\textcolor{red}{36}}\)
Therefore, \(6^{2/7}\) written in radical notation and simplified is \(\boldsymbol{\sqrt[7]{36}}\).
ii. We can rewrite this in exponential notation by placing the root (\(2\)) as the denominator and the power (\(1\)) as the numerator. Then, we can simplify:
\(= \textcolor{red}{64}^{\textcolor{green}{1}/\textcolor{blue}{2}}\)
\(= \textcolor{red}{8}\)
Therefore, \(\sqrt{64}\) written in radical form is \(64^{1/2}\) and simplified is \(\boldsymbol{8}\).
\(\sqrt[3]{3a^{5}}\)
We can rewrite this is exponential notation by placing the root (\(3\)) as the denominator and the the power (\(5\)) as the numerator. Then, we can simplify:
\(= (\textcolor{red}{3a})^{\textcolor{green}{5}/\textcolor{blue}{3}}\)
\(= \textcolor{red}{3}^{\textcolor{green}{5}/\textcolor{blue}{3}}\textcolor{red}{a}^{\textcolor{green}{5}/\textcolor{blue}{3}}\)
Therefore, \(\sqrt[3]{3a^{5}}\) in exponential notation can be written as either \(\boldsymbol{(3a)^{5/3}}\) or \(\boldsymbol{3^{5/3}a^{5/3}}\).
\((81x)^{-3/4}\)
We can rewrite this in radical notation by placing the numerator (\(-3\)) as the power and the denominator (\(4\)) as the root:
We can now simplify the expression by raising the coefficient and variable to the exponent:
\(= (\textcolor{red}{3x}^{1/4})^{-3}\)
\(= \cfrac{1}{(\textcolor{red}{3x}^{1/\textcolor{blue}{4}})^{\textcolor{green}{3}}}\)
\(= \cfrac{1}{\textcolor{red}{3}^{\textcolor{green}{3}}\textcolor{red}{x}^{(\textcolor{green}{3})(1/\textcolor{blue}{4})}}\)
\(= \cfrac{1}{\textcolor{red}{27x}^{\textcolor{green}{3}/\textcolor{blue}{4}}}\)
Therefore, we can determine that \((81x)^{-3/4}\) written in radical form is \(\boldsymbol{\sqrt[4]{({81x})^{-3}}}\) and simplified is \(\boldsymbol{\cfrac{1}{27x^{3/4}}}\).
\(-125^{1/3}\)
We can rewrite this in radical notation by placing the numerator (\(1\)) as the power and the denominator (\(3\)) as the root. Note where we place the negative sign:
\(= -\sqrt[\textcolor{blue}{3}]{\textcolor{red}{125}^{\textcolor{green}{1}}}\)
\(= \textcolor{red}{-5}\)
Therefore, we can determine that \(-125^{1/3}\) written in exponential notation is \( \boldsymbol{-\sqrt[3]{125^{1}}}\) and simplified is \(\boldsymbol{-5}\).
Simplify the following. Keep answers as exact reduced fractions and don't leave answers with negative exponents:
i. We cannot simplify the expression inside the brackets any further. Instead, we can multiply everything inside the bracket by its shared exponent:
\(= (256^\textcolor{red}{1}a^{\textcolor{red}{12}}b^{\textcolor{red}{20}})^{\textcolor{red}{3/4}}\)
\(= 256^\textcolor{red}{3/4}a^{\textcolor{red}{9}}b^{\textcolor{red}{15}}\)
We can simplify further by using Fractional Expnent Law on \(256^{3/4}\):
\(= \textcolor{red}{256^{3/4}}a^9b^15\)
\(= \sqrt[\textcolor{red}{4}]{\textcolor{red}{256^3}}a^9b^15\)
\(= \textcolor{red}{64}a^9b^15\)
Therefore, we can determine that \((256a^{12}b^{20})^{3/4}\) simplified is \(\boldsymbol{64a^9b^15}\).
ii. First, we can simplify the expression by dividing the coefficients:
\(= \cfrac{\textcolor{red}{25}x^{1/3}}{\textcolor{red}{5}x^{1/4}}\)
\(= \cfrac{\textcolor{red}{25}}{\textcolor{red}{5}}\cdot\cfrac{x^{1/3}}{x^{1/4}}\)
\(= \textcolor{red}{5}\cfrac{x^{1/3}}{x^{1/4}}\)
Next, we can further simplify the expression by subtracting like terms. We can do this by giving both variables a common denominator (in this case, \(12\)):
\(= 5\cfrac{x^{\textcolor{red}{1/3}}}{x^{\textcolor{red}{1/4}}}\)
\(= 5\cfrac{x^{\textcolor{red}{4/12}}}{x^{\textcolor{red}{3/12}}}\)
\(= 5x^{\textcolor{red}{1/12}}\)
Therefore, we can determine that \(\cfrac{25x^{1/3}}{5x{1/4}}\) simplified is \(\boldsymbol{5x^{1/12}}\).
\((8x⁶y⁹)^{1/3}(27x⁻¹²y⁶)^{-1/3}\)
First, we can raise both parts of the expression to their respective exponents. Then, we can simplify the expression by adding and subtracting like terms:
\(= (8^\textcolor{red}{1}x\textcolor{red}{⁶}y\textcolor{red}{⁹})^{\textcolor{red}{1/3}}(27^\textcolor{green}{1}x\textcolor{green}{⁻¹²}y\textcolor{green}{⁶})^{\textcolor{green}{-1/3}}\)
\(= (8^\textcolor{red}{1/3}x^\textcolor{red}{2}y^\textcolor{red}{3})(27^\textcolor{green}{-1/3}x^\textcolor{green}{{4}}y^{\textcolor{green}{-2}})\)
We can then raise the numeric coefficients (\(8\) and \(27\)) to their respective coefficients:
\(= (\textcolor{red}{8^{1/3}}x^2y^3)(\textcolor{red}{27^{-1/3}}x^{4}y^{-2})\)
\(= (\sqrt[\textcolor{red}{3}]{\textcolor{red}{8}^{1}}x^2y^3)(\sqrt[3]{\textcolor{red}{27^{-1}}}x^{4}y^{-2})\)
\(= {(\textcolor{red}{2}}x^2y^3)({\textcolor{red}{3^{-1}}}x^{4}y^{-2})\)
Next, we can simplify the expression by adding like exponents and shifting values with negative exponents to the opposite side:
\(= {(\textcolor{red}{2}}x^\textcolor{green}{2}y^\textcolor{blue}{3})({\textcolor{red}{3^{-1}}}x^{\textcolor{green}{4}}y^{\textcolor{blue}{-2}})\)
\(= \cfrac{\textcolor{red}{2}x^\textcolor{green}{6}y^\textcolor{blue}{1}}{{\textcolor{red}{3}}}\)
Therefore, we can determine that \((8x⁶y⁹)^{1/3}(27x⁻¹²y⁶)^{-1/3}\) simplified is \( \boldsymbol{\cfrac{2x^6y}{3}}\).
\(\left(\cfrac{64m¹⁵}{343}\right)^{-2/3}\)
First, we can raise the full expression to its shared exponent:
\(= \left(\cfrac{64^\textcolor{red}{1}m\textcolor{red}{¹⁵}}{343^\textcolor{red}{1}}\right)^{\textcolor{red}{-2/3}}\)
\(= \cfrac{64^\textcolor{red}{-2/3}m^\textcolor{red}{-10}}{343^\textcolor{red}{-2/3}}\)
Next, we can simplify the expression by raising \(64\) to its exponent:
\(= \cfrac{\textcolor{red}{64^{-2/3}}m^{-10}}{343^{-2/3}}\)
\(= \cfrac{(\sqrt[\textcolor{red}{3}]{\textcolor{red}{64^2}})^{\textcolor{red}{-1}}m^{-10}}{343^{-2/3}}\)
\(= \cfrac{\textcolor{red}{16^{-1}}m^{-10}}{343^{-2/3}}\)
We can further simplify by moving all values with negative exponents to the opposite side:
\(= \cfrac{\textcolor{red}{16^{-1}}\textcolor{green}{m^{-10}}}{\textcolor{blue}{343^{-2/3}}}\)
\(= \cfrac{\textcolor{blue}{343^{2/3}}}{\textcolor{red}{16^{1}}\textcolor{green}{m^{10}}}\)
Finally, we can raise \(343\) to its exponent:
\(= \cfrac{\textcolor{red}{343^{2/3}}}{16m^{10}}\)
\(= \cfrac{\sqrt[\textcolor{red}{3}]{\textcolor{red}{343^2}}}{16m^{10}}\)
\(= \cfrac{\textcolor{red}{49}}{16m^{10}}\)
Therefore, we can determine that \(\left(\cfrac{64m¹⁵}{343}\right)^{-2/3}\) simplified is \(\boldsymbol{\cfrac{49}{16m^{10}}}\).