10.5: Radicals with Mixed Indices
- Page ID
- 45138
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Knowing that a radical has the same properties as exponents (written as a ratio) allows us to manipulate radicals in new ways. One thing we are allowed to do is reduce, not just the radicand, but the index as well. Let’s take a look at a simple example.
Rewrite \(\sqrt[8]{x^6y^2}\) as a reduced radical with root \(4\).
Solution
We can rewrite the radical in its rational exponent form, then reduce each exponent fraction.
\[\begin{array}{rl}\sqrt[8]{x^6y^2}&\text{Rewrite the root }8\text{ as a rational exponent} \\ (x^6y^2)^{\dfrac{1}{8}}&\text{Multiply exponents} \\ x^{\dfrac{6}{8}}y^{\dfrac{2}{8}}&\text{Reduce each exponent fraction} \\ x^{\dfrac{3}{4}}y^{\dfrac{1}{4}}&\text{All exponents have denominator }4,\text{ rewrite in radical form} \\ \sqrt[4]{x^3y}&\text{Radical in reduced form with root }4\end{array}\nonumber\]
Reduce Radicals
Notice we reduced the index by dividing the index and all exponents in the radicand by the same number, e.g., \(2\) in Example 10.5.1. If we notice a common factor between the index and all exponents of every factor in the radicand, then we can reduce the radical by dividing by that common factor.
If given a radical with root \(m\cdot n\) and radicand \(a^{mp}\), then \[\sqrt[mn]{a^{mp}}=\sqrt[\cancel{m}n]{a^{\cancel{m}p}}=\sqrt[n]{a^p}\nonumber\]
Reduce: \(\sqrt[24]{a^6b^9c^{15}}\)
Solution
We can rewrite the radical with the root and exponents in the radicand as a product with a common factor, then reduce the radical.
\[\begin{array}{rl}\sqrt[24]{a^6b^9c^{15}}&\text{Rewrite root and each exponent as a product with the common factor }3 \\ \sqrt[\color{blue}{3}\color{black}{\cdot 8}]{a^{\color{blue}{3}\color{black}{\cdot 2}}b^{\color{blue}{3}\color{black}{\cdot 3}}c^{\color{blue}{3}\color{black}{\cdot 5}}}&\text{Reduce by a common factor of }3 \\ \sqrt[\color{blue}{\cancel{3}}\color{black}{\cdot 8}]{a^{\color{blue}{\cancel{3}}\color{black}{\cdot 2}}b^{\color{blue}{\cancel{3}}\color{black}{\cdot 3}}c^{\color{blue}{\cancel{3}}\color{black}{\cdot 5}}}&\text{Simplify} \\ \sqrt[8]{a^2b^3c^5}&\text{Radical in reduced form with root }8\end{array}\nonumber\]
We can use the same process even if there are coefficients in the radicand. We just have to rewrite the coefficient with an exponent that includes the common factor of the exponents, and then reduce the radical as usual.
Reduce: \(\sqrt[9]{8m^6n^3}\)
Solution
First, we’ll need to rewrite the coefficient \(8\) with an exponent that includes the common factor of the exponents. Then we can reduce the radical as usual.
\[\begin{array}{rl}\sqrt[9]{8m^6n^3}&\text{Rewrite coefficient }8\text{ with an exponent including the common factor }3 \\ \sqrt[9]{2^{\color{blue}{3}}\color{black}{m^6n^3}}&\text{Rewrite root and each exponent as a product with the common factor }3 \\ \sqrt[\color{blue}{3}\color{black}{\cdot 3}]{2^{\color{blue}{3}\color{black}{\cdot 1}}m^{\color{blue}{3}\color{black}{\cdot 2}}n^{\color{blue}{3}\color{black}{\cdot 1}}}&\text{Reduce by a common factor of }3 \\ \sqrt[\color{blue}{\cancel{3}}\color{black}{\cdot 3}]{2^{\color{blue}{\cancel{3}}\color{black}{\cdot 1}}m^{\color{blue}{\cancel{3}}\color{black}{\cdot 2}}n^{\color{blue}{\cancel{3}}\color{black}{\cdot 1}}}&\text{Simplify} \\ \sqrt[3]{2m^2n}&\text{Radical in reduced form with root }3\end{array}\nonumber\]
Multiply Radicals with Different Indices
We can apply the method of reducing radicals to multiply radicals with different indices. Let’s consider an example using rational exponents, then identify a pattern.
Multiply: \(\sqrt[3]{ab^2}\cdot\sqrt[4]{a^2b}\)
Solution
We can rewrite the radicals in its rational exponent form, find a common denominator, then reduce each exponent fraction.
\[\begin{array}{rl}\sqrt[3]{ab^2}\sqrt[4]{a^2b}&\text{Rewrite as rational exponents} \\ (ab^2)^{\dfrac{1}{3}}(a^2b)^{\dfrac{1}{4}}&\text{Multiply exponents} \\ a^{\dfrac{1}{3}}b^{\dfrac{2}{3}}a^{\dfrac{2}{4}}b^{\dfrac{1}{4}}&\text{Rewrite each exponent with common denominator }12 \\ a^{\dfrac{4}{\color{blue}{12}}}b^{\dfrac{8}{\color{blue}{12}}}a^{\dfrac{6}{\color{blue}{12}}}b^{\dfrac{3}{\color{blue}{12}}}&\text{Rewrite in radical form with index }12 \\ \sqrt[\color{blue}{12}]{\color{red}{a^4}\color{black}{\cdot b^8\cdot }\color{red}{a^6}\color{black}{\cdot b^3}}&\text{Add exponents with same base} \\ \sqrt[12]{a^{10}b^{11}}&\text{Produce with common root }12\end{array}\nonumber\]
To multiply radicals with different indices, we need to find a common denominator, which is the lowest common multiple (LCM) between the roots. Once we obtain the LCM, we can multiply each root and exponent in the radicand to obtain the LCM, and rewrite as one radical.
Let \(n\), \(p\), \(m\) be positive nonzero integers, and the lowest common multiple be \(m\), i.e., \(LCM(n, p) = m\), then
\[\sqrt[n]{a}\cdot\sqrt[p]{b}=\sqrt[m]{a^r}\cdot\sqrt[m]{b^t}=\sqrt[m]{a^rb^t},\nonumber\]
where the exponents \(r=\dfrac{m}{n}\) and \(t=\dfrac{m}{p}\).
Multiply: \(\sqrt[4]{a^2b^3}\cdot\sqrt[6]{a^2b}\)
Solution
Let’s find the \(LCM(4, 6)\) and rewrite each radical with the LCM. Then write as one radical.
\[\begin{array}{rl}\sqrt[4]{a^2b^3}\cdot\sqrt[6]{a^2b}&\text{Rewrite radicals with LCM }12 \\ \sqrt[\color{blue}{3}\color{black}{\cdot 4}]{a^{\color{blue}{3}\color{black}{\cdot 2}}b^{\color{blue}{3}\color{black}{\cdot 3}}}\cdot\sqrt[\color{blue}{2}\color{black}{\cdot 6}]{a^{\color{blue}{2}\color{black}{\cdot 2}}b^{\color{blue}{2}\color{black}{\cdot 1}}}&\text{Multiply }3\text{ through first radical and multiply }2\text{ through second radical} \\ \sqrt[12]{a^6b^9}\cdot\sqrt[12]{a^4b^2}&\text{Simplify and write as one radical with root }12 \\ \sqrt[12]{a^6b^9 \cdot a^4b^2}&\text{Add exponents with same base} \\ \sqrt[12]{a^{10}b^{11}}&\text{Product with common root }12\end{array}\nonumber\]
Multiply: \(\sqrt[5]{x^3y^4}\cdot\sqrt[3]{x^2y}\)
Solution
Let’s find the \(LCM(3, 5)\) and rewrite each radical with the LCM. Then write as one radical.
\[\begin{array}{rl}\sqrt[5]{x^3y^4}\cdot\sqrt[3]{x^2y}&\text{Rewrite radicals with LCM }15 \\ \sqrt[\color{blue}{3}\color{black}{\cdot 5}]{x^{\color{blue}{3}\color{black}{\cdot 3}}y^{\color{blue}{3}\color{black}{\cdot 4}}}\cdot\sqrt[\color{blue}{5}\color{black}{\cdot 3}]{x^{\color{blue}{5}\color{black}{\cdot 2}}y^{\color{blue}{5}\color{black}{\cdot 1}}}&\text{Multiply }3\text{ through first radical and multiply }5\text{ through second radical} \\ \sqrt[15]{x^9y^{12}}\cdot\sqrt[15]{x^{10}y^5}&\text{Simplify and write as one radical with root }15 \\ \sqrt[15]{x^9y^{12}\cdot x^{10}y^5}&\text{Add exponents with same base} \\ \sqrt[15]{x^{19}y^{17}}&\text{Simplify by extracting out one factor of }x\text{ and }y \\ xy\sqrt[15]{x^4y^2}&\text{Product with common root }15\text{ and extracted factors }x\text{ and }y\end{array}\nonumber\]
Multiply: \(\sqrt{3x(y+x)}\cdot\sqrt[3]{9x(y+z)^2}\)
Solution
Let’s find the \(LCM(2, 3)\) and rewrite each radical with the LCM. Then write as one radical. Note, even though there is a binomials in each radicand, the method stays the same. Recall, methods never change, only problems.
\[\begin{array}{rl}\sqrt{3x(y+z)}\cdot\sqrt[3]{9x(y+z)^2}&\text{Rewrite radicals with LCM }6 \\ \sqrt[\color{blue}{3}\color{black}{\cdot 2}]{3^{\color{blue}{3}\color{black}{\cdot 1}}x^{\color{blue}{3}\color{black}{\cdot 1}}(y+z)^{\color{blue}{3}\color{black}{\cdot 1}}}\cdot\sqrt[\color{blue}{2}\color{black}{\cdot 3}]{3^{\color{blue}{2}\color{black}{\cdot 2}}x^{\color{blue}{2}\color{black}{\cdot 1}}(y+z)^{\color{blue}{2}\color{black}{\cdot 2}}} &\text{Multiply }3\text{ through first radical and multiply }2 \\ &\text{through second radical} \\ \sqrt[6]{3^3x^3(y+z)^3}\cdot\sqrt[6]{3^4x^2(y+z)^4}&\text{Simplify and write as one radical with root }6 \\ \sqrt[6]{3^3x^3(y+z)^3\cdot 3^4x^2(y+z)^4}&\text{Add exponents with same base} \\ \sqrt[6]{3^7x^5(y+z)^7}&\text{Simplify by reducing out one factor of }3\text{ and }(y+z) \\ 3(y+z)\sqrt[6]{3x^5(y+z)}&\text{Product with common root }6\text{ and extracted factors} \\ &3\text{ and }(y+z)\end{array}\nonumber\]
Originally, the radical was just a check mark with the rest of the radical expression in parenthesis. In 1637, Rene Descartes was the first to put a line over the entire radical expression.
Divide Radicals with Different Indices
Luckily, the same process is used for dividing radicals with mixed indices as we used multiplying radicals with mixed indices. Since the final expression cannot have radicals in the denominator, then there may be an additional step of rationalizing the denominator.
Divide: \(\dfrac{\sqrt[6]{x^4y^3z^2}}{\sqrt[8]{x^7y^2z}}\)
Solution
Let’s find the \(LCM(6, 8)\) and rewrite each radical with the LCM. Then write as one radical. Note, even though we are simplifying a quotient, we still rationalize the denominator when necessary.
\[\begin{array}{rl}\dfrac{\sqrt[6]{x^4y^3z^2}}{\sqrt[8]{x^7y^2z}}&\text{Rewrite radicals with LCM }24 \\ \dfrac{\sqrt[\color{blue}{4}\color{black}{\cdot 6}]{x^{\color{blue}{4}\color{black}{\cdot 4}}y^{\color{blue}{4}\color{black}{\cdot 3}}z^{\color{blue}{4}\color{black}{\cdot 2}}}}{\sqrt[\color{blue}{3}\color{black}{\cdot 8}]{x^{\color{blue}{3}\color{black}{\cdot 7}}y^{\color{blue}{3}\color{black}{\cdot 2}}z^{\color{blue}{3}\color{black}{\cdot 1}}}}&\text{Multiply }4\text{ through numerator radical and multiply }3\text{ through denominator radical} \\ \dfrac{\sqrt[24]{x^{16}y^{12}z^8}}{\sqrt[24]{x^{21}y^6z^3}}&\text{Simplify and write as one radical with root }24 \\ \sqrt[24]{\dfrac{x^{16}y^{12}z^{8}}{x^{21}y^6z^3}}&\text{Reduce factors with same base} \\ \sqrt[24]{\dfrac{y^6z^5}{x^5}}&\text{Rationalize the denominator} \\ \dfrac{\sqrt[24]{y^6z^5}}{\sqrt[24]{x^5}}\cdot\dfrac{\sqrt[24]{x^{19}}}{\sqrt[24]{x^{19}}}&\text{Multiply numerator and denominator by }\sqrt[24]{x^{19}} \\ \dfrac{\sqrt[24]{x^{19}y^6z^5}}{\sqrt[24]{x^{24}}}&\text{Simplify} \\ \dfrac{\sqrt[24]{x^{19}y^6z^5}}{x}&\text{Quotient with common root }24\text{ and rationalized denominator}\end{array}\nonumber\]
Radicals with Mixed Indices Homework
Reduce the following radicals.
\(\sqrt[8]{16x^4y^6}\)
\(\sqrt[12]{64x^4y^6z^8}\)
\(\sqrt[6]{\dfrac{16x^2}{9y^4}}\)
\(\sqrt[12]{x^6y^9}\)
\(\sqrt[8]{x^6y^4z^2}\)
\(\sqrt[9]{8x^3y^6}\)
\(\sqrt[4]{9x^2y^6}\)
\(\sqrt[4]{\dfrac{25x^3}{16x^5}}\)
\(\sqrt[15]{x^9y^{12}z^6}\)
\(\sqrt[10]{64x^8y^4}\)
\(\sqrt[4]{25y^2}\)
\(\sqrt[16]{81x^8y^{12}}\)
Multiply or divide and simplify completely.
\(\sqrt[3]{5}\cdot\sqrt{6}\)
\(\sqrt{x}\cdot\sqrt[3]{7y}\)
\(\sqrt{x}\cdot\sqrt[3]{x-2}\)
\(\sqrt[5]{x^2y}\cdot\sqrt{xy}\)
\(\sqrt[4]{xy^2}\cdot\sqrt[3]{x^2y}\)
\(\sqrt[4]{a^2bc^2}\cdot\sqrt[5]{a^2b^3c}\)
\(\sqrt{a}\cdot\sqrt[4]{a^3}\)
\(\sqrt[5]{b^2}\cdot\sqrt{b^3}\)
\(\sqrt{xy^3}\cdot\sqrt[3]{x^2y}\)
\(\sqrt[4]{9ab^3}\cdot\sqrt{3a^4b}\)
\(\sqrt[3]{3xy^2z}\cdot\sqrt[4]{9x^3yz^2}\)
\(\sqrt{27a^5(b+1)}\cdot\sqrt[3]{81a(b+1)^4}\)
\(\dfrac{\sqrt[3]{a^2}}{\sqrt[4]{a}}\)
\(\dfrac{\sqrt[4]{x^2y^3}}{\sqrt[3]{xy}}\)
\(\dfrac{\sqrt{ab^3c}}{\sqrt[5]{a^2b^3c^{-1}}}\)
\(\dfrac{\sqrt[4]{(3x-1)^3}}{\sqrt[5]{(3x-1)^3}}\)
\(\dfrac{\sqrt[3]{(2x+1)^2}}{\sqrt[5]{(2x+1)^2}}\)
\(\sqrt[3]{7}\cdot\sqrt[4]{5}\)
\(\sqrt[3]{y}\cdot\sqrt[5]{3z}\)
\(\sqrt[4]{3x}\cdot\sqrt{y+4}\)
\(\sqrt{ab}\cdot\sqrt[5]{2a^2b^2}\)
\(\sqrt[5]{a^2b^3}\cdot\sqrt[4]{a^2b}\)
\(\sqrt[6]{x^2yz^3}\cdot\sqrt[5]{x^2yz^2}\)
\(\sqrt[3]{x^2}\cdot\sqrt[6]{x^5}\)
\(\sqrt[4]{a^3}\cdot\sqrt[3]{a^2}\)
\(\sqrt[5]{a^3b}\cdot\sqrt{ab}\)
\(\sqrt{2x^3y^3}\cdot\sqrt[3]{4xy^2}\)
\(\sqrt{a^4b^3c^4}\cdot\sqrt[3]{ab^2c}\)
\(\sqrt{8x(y+z)^5}\cdot\sqrt[3]{4x^2(y+z)^2}\)
\(\dfrac{\sqrt[3]{x^2}}{\sqrt[5]{x}}\)
\(\dfrac{\sqrt[5]{a^4b^2}}{\sqrt[3]{ab^2}}\)
\(\dfrac{\sqrt[5]{x^3y^4z^9}}{\sqrt{xy^{-2}z}}\)
\(\dfrac{\sqrt[3]{(2+5x)^2}}{\sqrt[4]{(2+5x)}}\)
\(\dfrac{\sqrt[4]{(5-3x)^3}}{\sqrt[3]{(5-3x)^2}}\)