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This is one of our many quizzes on Simplifying exponential expressions. Click on this to access more such competitive quizzes on Algebraic Expressions.

Simplifying exponential expressions Quiz 2

mathematics

 Simplify $\ln \left( {{3^{x - 5}}} \right) = \ln \left( 6 \right)$ 

 Evaluate the exponent term and apply the $ln$ values to get the value of $x$.

 Given, $\ln \left( {{3^{x - 5}}} \right) = \ln \left( 6 \right) \\ \\ $ Evaluate the exponent term as: $ \\ \Rightarrow \left( {x - 5} \right)\ln \left( 3 \right) = \ln \left( 6 \right) \\ \Rightarrow \left( {x - 5} \right) = \dfrac{{\ln \left( 6 \right)}}{{\ln \left( 3 \right)}} \\ \Rightarrow x = \dfrac{{\ln \left( 6 \right)}}{{\ln \left( 3 \right)}} + 5 \\ $ W.K.T the value of $\ln \left( 6 \right) = 1.791$ and $\ln \left( 3 \right) = 1.098$, hence substituting it we get: $ \\  \Rightarrow x = \dfrac{{1.791}}{{1.098}} + 5 \\ \Rightarrow x = 1.631 + 5 \\ \Rightarrow x = 6.631$

 Evaluate the given expression $\dfrac{{{2^4}{x^2}{y^5}}}{{8{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}}$

 To evaluate the given expression simplify based on its power value of the given exponent.

 Given, $\dfrac{{{2^4}{x^2}{y^5}}}{{8{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}} \\ \\ $ We need to evaluate with respect to their powers i.e., $ \\ \Rightarrow \dfrac{{{2^4}{x^2}{y^5}}}{{8{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}} = \dfrac{{16{x^2}{y^5}}}{{8{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}} \\ $ Now, combine and simplify the like terms as: $ \\ \Rightarrow  \dfrac{{2{x^2}{y^5}}}{{{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}} \\ $ As, the denominator consists of like terms, we get: $ \\ \Rightarrow  \dfrac{{2{x^2}{y^5}}}{{{x^1}{y^2}{z^{ - 2}}}} \\ $ Simplifying numerator and denominator terms, we get: $ \\ \Rightarrow  2x{y^3}{z^2} \\ \therefore \Rightarrow \dfrac{{{2^4}{x^2}{y^5}}}{{8{x^2}{y^2}{x^{ - 1}}{z^{ - 2}}}} = 2x{y^3}{z^2}$

 Get the final expression of the given equation $\left( {{a^3}} \right)\left( {{a^2}} \right) + {\left( {2{b^2}} \right)^3}$

 To simplify the given expression apply the rule for multiplying two exponential expressions.

 Given, $\left( {{a^3}} \right)\left( {{a^2}} \right) + {\left( {2{b^2}} \right)^3} \\ \\ $ Here to simplify the expression, apply the rule for multiplying two exponential expressions with the same base: $ \\  \Rightarrow \left( {{a^3}} \right)\left( {{a^2}} \right) = {a^5} \\ $ Now multiplying two exponential numbers with exponents as: $ \\ \Rightarrow {\left( {2{b^2}} \right)^3} = {\left( 2 \right)^3} \cdot {\left( {{b^2}} \right)^3} \\ $ We know that, ${2^3} = 2 \cdot 2 \cdot 2 = 8$, apply rule of raising a power to an exponent as: $ \\ \Rightarrow {\left( {2{b^2}} \right)^3} = 8{b^6} \\ $ Therefore, combining both the terms, we get: $ \\ \Rightarrow {a^5} + 8{b^6}$

 Apply log on both sides of the given expression and then simplify the terms to get the value of $x$.

 Given, ${5^{2x}} = 21 \\ \\ $ Apply log on both sides of the equation as: $ \\ \Rightarrow {\log _{10}}\left( {{5^{2x}}} \right) = {\log _{10}}\left( {21} \right) \\ \Rightarrow 2x{\log _{10}}\left( 5 \right) = {\log _{10}}\left( {21} \right) \\ \Rightarrow 2x = \dfrac{{{{\log }_{10}}\left( {21} \right)}}{{{{\log }_{10}}\left( 5 \right)}} \\ \Rightarrow 2x = \dfrac{{1.322}}{{0.698}} \\ \Rightarrow 2x = 1.894 \\ \Rightarrow x = \dfrac{{1.894}}{2} \\ \Rightarrow x = 0.946991 \\ \Rightarrow x \approx 0.95$

 Simplify $\sqrt[{2x + 4}]{{{4^{x + 8}}}} = \sqrt[6]{{128}}$

 Apply power rule and then simplify the terms to get the value of $x$.

 Given, $\sqrt[{2x + 4}]{{{4^{x + 8}}}} = \sqrt[6]{{128}} \\ \\ $ With respect to Power rule, we get: $ \\ \Rightarrow {4^{\frac{{x + 8}}{{2x + 4}}}} = {2^{\frac{7}{6}}} \\ $ W.K.T, ${2^2} = 4$, hence; $ \\ \Rightarrow {2^2}^{\frac{{x + 8}}{{2x + 4}}} = {2^{\frac{7}{6}}} \\ \Rightarrow {2^{\frac{{x + 8}}{{x + 4}}}} = {2^{\frac{7}{6}}} \\ $ As, $2$ is common, simplifying we get: $ \\ \Rightarrow \dfrac{{x + 8}}{{x + 4}} = \dfrac{7}{6} \\ \Rightarrow 6\left( {x + 8} \right) = 7\left( {x + 4} \right) \\ \Rightarrow 6x + 48 = 7x + 14 \\ $ Combine all the common terms, and simplify the terms as: $ \\ \Rightarrow 6x - 7x =  + 14 - 48 \\ \Rightarrow  - x =  - 34$