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This is the second of our many quizzes on exponents. Click on this to access more such competitive quizzes on exponents.

Exponents Quiz 2

mathematics

 Evaluate ${\left( {\sqrt {729} } \right)^{\frac{1}{3}}} \times {\left( {\sqrt {15625} } \right)^{\frac{1}{3}}} + \dfrac{1}{{{{729}^{ - \frac{1}{6}}}}} - {2187^{\frac{1}{7}}} $.

 Use the identity ${a^{ - m}} = \dfrac{1}{{{a^m}}};{a^m} \times {a^n} = {a^{m + n}}$ to solve the expression

  $\text{Given equation can be written as }\\{\left( {27} \right)^{\frac{1}{3}}} \times {\left( {125} \right)^{\frac{1}{3}}} + {729^{\frac{1}{6}}} - {2187^{\frac{1}{7}}}  \\ \Rightarrow  {\left( {{3^3}} \right)^{\frac{1}{3}}} \times {\left( {{5^3}} \right)^{\frac{1}{3}}} + {\left( {{3^6}} \right)^{\frac{1}{6}}} - {\left( {{3^7}} \right)^{\frac{1}{7}}}  \\ \Rightarrow  3 \times 5 + 3 - 3  \Rightarrow15$ 

 Evaluate ${\left( {{9^3}} \right)^{\frac{1}{6}}} - {\left( {{{144}^3}} \right)^{\frac{1}{6}}} + {\left( {{{64}^2}} \right)^{\frac{1}{6}}} + {\left( {{{216}^2}} \right)^{\frac{1}{6}}}$

  $\text{Given equation can be written as }\\ {\left( 9 \right)^{3 \times \frac{1}{6}}} - {\left( {144} \right)^{3 \times \frac{1}{6}}} + {\left( {64} \right)^{2 \times \frac{1}{6}}} + {\left( {216} \right)^{2 \times \frac{1}{6}}} \\ \Rightarrow {\left( 9 \right)^{\frac{1}{2}}} - {\left( {144} \right)^{\frac{1}{2}}} + {\left( {64} \right)^{\frac{1}{3}}} + {\left( {216} \right)^{\frac{1}{3}}} \\ \Rightarrow 3 - 12 + 4 + 6  \Rightarrow1$ 

 Evaluate $\left( {{5^{2m + n}}} \right) \times \left( {{5^{2m - n}}} \right) \times \dfrac{1}{{\left( {{5^{ - 2m - 2n}}} \right)}} $

 $\text{Given equation can be written as }{5^{2m}} \times {5^n} \times \dfrac{{{5^{2m}}}}{{{5^n}}} \times {5^{2m + 2n}}  \\ \Rightarrow  {5^{2m}} \times {5^n} \times \dfrac{{{5^{2m}}}}{{{5^n}}} \times {5^{2m}} \times {5^{2n}}  \\ \Rightarrow {5^{6m}} \times {5^{2n}} \\ \Rightarrow {5^{6m + 2n}}$

 Evaluate ${6561^{ - \frac{1}{8}}} \times \dfrac{1}{{{{2187}^{ - \frac{1}{7}}}}} \times {729^{ - \frac{1}{3}}} \times \dfrac{1}{{{{324}^{ - \frac{1}{2}}}}} $

 $\text{Given equation can be written as }\dfrac{1}{{{{6561}^{\frac{1}{8}}}}} \times {2187^{\frac{1}{7}}} \times \dfrac{1}{{{{729}^{\frac{1}{3}}}}} \times {324^{\frac{1}{2}}}  \\ \Rightarrow \dfrac{1}{{{{\left( {{3^8}} \right)}^{\frac{1}{8}}}}} \times {\left( {{3^7}} \right)^{\frac{1}{7}}} \times \dfrac{1}{{{{\left( {{9^3}} \right)}^{\frac{1}{3}}}}} \times {\left( {{{18}^2}} \right)^{\frac{1}{2}}}  \\ \Rightarrow \dfrac{1}{3} \times 3 \times \dfrac{1}{9} \times 18  \Rightarrow2$

 Evaluate ${\left( {{{\left( {81} \right)}^2}} \right)^{\frac{1}{8}}} \times \dfrac{1}{{{{\left( {{{16}^2}} \right)}^{ - \frac{1}{4}}}}} \times {1728^{ - \frac{1}{3}}} \times \dfrac{1}{{{{324}^{ - \frac{1}{2}}}}} $

 $\text{Given equation can be written as }\\{\left( {81} \right)^{2 \times \frac{1}{8}}} \times \dfrac{1}{{{{\left( {16} \right)}^{ - 2 \times \frac{1}{4}}}}} \times {324^{\frac{1}{2}}} \times \dfrac{1}{{{{1728}^{\frac{1}{3}}}}}  \\ \Rightarrow  {\left( {81} \right)^{\frac{1}{4}}} \times \dfrac{1}{{{{\left( {16} \right)}^{ - \frac{1}{2}}}}} \times {\left( {{{18}^2}} \right)^{\frac{1}{2}}} \times \dfrac{1}{{{{\left( {{{12}^3}} \right)}^{\frac{1}{3}}}}}  \\ \Rightarrow  {\left( {{3^4}} \right)^{\frac{1}{4}}} \times {\left( {16} \right)^{\frac{1}{2}}} \times 18 \times \dfrac{1}{{12}}  \\ \Rightarrow  3 \times 4 \times 18 \times \dfrac{1}{{12}} \Rightarrow18$