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This is one of our many quizzes on Simplifying higher roots. Click on this to access more such competitive quizzes on Number System.

Simplifying higher roots Quiz 2

mathematics

 Find the value of $B$ if $B=ab$ , where $a=\sqrt[5]{{{\left( 64 \right)}^{2}}}$ and $b=\sqrt[4]{{{\left( 81 \right)}^{2}}}$.

 Simplify the given roots using the formula $\sqrt[n]{a}={{a}^{\frac{1}{n}}}$ and then solve the problem to get the final answer.

  It is given to find the value of $B$ where $B$ is the product of $a,b$ $ \\ $ Let us find the values of $a$ and $b$ separately. $ \\ a=\sqrt[5]{{{\left( 64 \right)}^{2}}} \\ \Rightarrow a={{\left( {{\left( {{\left( 2 \right)}^{5}} \right)}^{2}} \right)}^{\frac{1}{5}}} \\ \Rightarrow a={{2}^{2}} \\ \Rightarrow a=4 \\ b=\sqrt[4]{{{\left( 81 \right)}^{2}}} \\  \Rightarrow b={{\left( {{\left( {{\left( {{3}^{2}} \right)}^{2}} \right)}^{2}} \right)}^{\frac{1}{4}}} \\ \Rightarrow b={{3}^{2}} \\ \Rightarrow b=9 \\ $ Therefore,  $B=ab=4\times 9 \\ \Rightarrow 36 $

 Evaluate the radical expression $\sqrt[4]{\text{C}}$ where $\text{C}=176{{m}^{3}}{{n}^{2}}$

 Check every term in the given equation for simplifying the higher roots and then simplify the equation.

  It is given to find the value of $\sqrt[4]{\text{C}}$ where $\text{C}$ is $176{{m}^{3}}{{n}^{2}}$. $ \\ $ Now let us find the value of $\sqrt[4]{\text{C}}$  $ \\ \Rightarrow \sqrt[4]{\text{C}}=\sqrt[4]{176{{m}^{3}}{{n}^{2}}} \\ $ Though we can’t simplify $\sqrt[4]{\text{176}}$. Let us have a look at the $\sqrt[4]{{{m}^{3}}{{n}^{2}}}$. $ \\ \Rightarrow \sqrt[4]{{{m}^{3}}{{n}^{2}}}=\sqrt{n}\sqrt[4]{{{m}^{3}}} \\ $ So, therefore the simplified equation is  $ \\ \sqrt[4]{\text{C}}=\sqrt[4]{176{{m}^{3}}}\sqrt{n}$

 Simplify $\sqrt[3]{243{{m}^{5}}}-\sqrt[3]{-108{{n}^{10}}}$. 

 To solve the given complex problem, divide them into the terms and then solve them using the exponent power formula and subtract the terms to get the required answer.

To simplify the given equation $\sqrt[3]{243{{m}^{5}}}-\sqrt[3]{-108{{n}^{10}}}$. Let us simplify the equation term wise. $ \\ $ Simplifying the first term $ \\ \sqrt[3]{243{{m}^{5}}}={{\left( 243{{m}^{5}} \right)}^{\frac{1}{3}}} \\ \Rightarrow \sqrt[3]{243{{m}^{5}}}={{\left( {{3}^{3}}{{m}^{3}} \right)}^{\frac{1}{3}}}{{\left( {{3}^{2}}{{m}^{2}} \right)}^{\frac{1}{3}}} \\ \Rightarrow \sqrt[3]{243{{m}^{5}}}=3m\sqrt[3]{{{\left( 3m \right)}^{2}}} \\ $ Now simplifying the second term  $ \\ \sqrt[3]{-108{{n}^{10}}}=\sqrt[3]{-{{3}^{3}}{{4}^{2}}{{n}^{10}}} \\ \Rightarrow \sqrt[3]{-108{{n}^{10}}}={{\left( -{{3}^{3}}{{4}^{2}}{{n}^{10}} \right)}^{\frac{1}{3}}} \\ \Rightarrow \sqrt[3]{-108{{n}^{10}}}=3n\sqrt[3]{-{{4}^{2}}{{n}^{7}}} \\ $ Therefore, the simplified form of given expression will be  $ \\ \therefore \sqrt[3]{243{{m}^{5}}}-\sqrt[3]{-108{{n}^{10}}}=3m\sqrt[3]{{{\left( 3m \right)}^{2}}}-3n\sqrt[3]{-{{4}^{2}}{{n}^{7}}}$

 If the simplified equation of the equation $\sqrt[4]{765{{r}^{12}}}-\sqrt[4]{968{{r}^{15}}}$ is ${{r}^{n}}\left( \text{a}-{{r}^{\frac{3}{4}}}\text{b} \right)$. Then, find the value of $n$?

 Simplify the given expression as far as possible and then compare with the second expression to find the value of $n$.

 Let us simplify the given expression $\sqrt[4]{765{{r}^{12}}}-\sqrt[4]{968{{r}^{15}}}$ and find the value of $n$. $ \\ x=\sqrt[4]{765{{r}^{12}}}-\sqrt[4]{968{{r}^{15}}} \\ $ Let us rewrite the equation by simplifying the powers $ \\ x={{\left( 765{{r}^{12}} \right)}^{\frac{1}{4}}}-{{\left( 968{{r}^{15}} \right)}^{\frac{1}{4}}} \\ \Rightarrow x={{r}^{3}}{{\left( 765 \right)}^{\frac{1}{4}}}-{{r}^{\frac{15}{4}}}{{\left( 968 \right)}^{\frac{1}{4}}} \\ $ The given simplified expression is, $ \\ x={{r}^{n}}\left( \text{a}-{{r}^{\frac{3}{4}}}\text{b} \right) \\ \Rightarrow x={{r}^{n}}\text{a}-{{r}^{n+\dfrac{3}{4}}}\text{b} \\ $ Now by comparing our expression with given expression the exponent of $r$ in second term is taken as, $ \\ n+\dfrac{3}{4}=\dfrac{15}{4} \\ \Rightarrow n=\dfrac{15}{4}-\dfrac{3}{4} \\ \Rightarrow n=3 \\ $ So, therefore the expression will be  $ \\ \therefore x={{r}^{3}}\left( {{\left( 765 \right)}^{\frac{1}{4}}}-{{r}^{\frac{3}{4}}}{{\left( 968 \right)}^{\frac{1}{4}}} \right)$

 Simplify $\sqrt[8]{1024{{m}^{3}}}+\sqrt[4]{177147{{q}^{5}}}-ab$. Where $a=\sqrt[6]{177147}$ and $b=\sqrt[4]{1024}$.

 Solve the given expression into 3 terms and then simplify them separately. Then add and subtract the required terms to get the simplified value of the given expression.

 For simplifying the given expression $\sqrt[8]{1024{{m}^{3}}}+\sqrt[4]{177147{{q}^{5}}}-ab$. Let us divide the expression in $3$ terms. $ \\ $ Let us solve the ${{1}^{st}}$ term. $ \\ \sqrt[8]{1024{{m}^{3}}}={{\left( {{2}^{10}}{{m}^{3}} \right)}^{\frac{1}{8}}} \\ \Rightarrow \sqrt[8]{1024{{m}^{3}}}=2\left( \sqrt[8]{{{2}^{2}}{{m}^{3}}} \right) \\ $ Let us solve the ${{2}^{nd}}$ term. $ \\ \sqrt[4]{177147{{q}^{5}}}={{\left( {{3}^{11}}{{q}^{5}} \right)}^{\frac{1}{4}}} \\ \Rightarrow \sqrt[4]{177147{{q}^{5}}}=3{{\left( {{3}^{7}}{{q}^{5}} \right)}^{\frac{1}{4}}} \\ \Rightarrow \sqrt[4]{177147{{q}^{5}}}=3\left( \sqrt[4]{{{3}^{7}}{{q}^{5}}} \right) \\ $ Let us start solving ${{3}^{rd}}$ term  $ \\ ab=\sqrt[6]{177147} \cdot \sqrt[4]{1024} \\ \Rightarrow ab=3\sqrt[6]{243}\sqrt[4]{1024} \\ \Rightarrow ab=3\cdot \:4\sqrt[6]{243}\sqrt{2} \\ \Rightarrow ab=12\sqrt[6]{243}\sqrt{2} \\ $ Therefore, by adding first $2$ terms and subtracting the ${{3}^{rd}}$ term we get $ \\ \therefore \sqrt[8]{1024{{m}^{3}}}+\sqrt[4]{177147{{q}^{5}}}-ab=2\left( \sqrt[8]{{{2}^{2}}{{m}^{3}}} \right)+3\left( \sqrt[4]{{{3}^{7}}{{q}^{5}}} \right)-12\sqrt[6]{243}\sqrt{2}$