Question

Solve the equation: \[{5^x}\sqrt[x]{{{8^{x - 1}}}} = 500\]. Find the rational root.

Answer 1

Hint: According to the question, To find the rational roots of the given equation we have to make the right hand side zero, for that we will simplify it and then divide it from the left hand side. We will also use the property of log for simplifying all the powers in the equation.

Formula used: Here we use the identity \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] and \[\log {n^m} = m*\log n\]

Complete step-by-step answer:
As the given equation is \[{5^x}*\sqrt[x]{{{8^{x - 1}}}} = 500\] .
Rewriting the equation as \[{5^x} + {8^{\left( {\dfrac{{x + 1}}{x}} \right)}} = 500\] .
Simplifying by rewriting 500 as \[{5^3}*{2^2}\] which is \[125*4\] that is equal to 500.
\[ \Rightarrow {5^x}*{8^{\left( {\dfrac{{x - 1}}{x}} \right)}} = {5^3}*{2^2}\]
Simplifying by rewriting 8 as \[{2^3}\]
\[ \Rightarrow {5^x}*{2^{3\left( {\dfrac{{x - 1}}{x}} \right)}} = {5^3}*{2^2}\]
Multiplying by 3 with the existing power of 2 on left hand side
\[ \Rightarrow {5^x}*{2^{\left( {\dfrac{{3x - 3}}{x}} \right)}} = {5^3}*{2^2}\]
Taking \[{5^3}*{2^2}\] on left side by dividing all the values of left hand side by \[{5^3}*{2^2}\] .
\[ \Rightarrow \dfrac{{\left( {{5^x}*{2^{\left( {\dfrac{{3x - 3}}{x}} \right)}}} \right)}}{{{5^3}*{2^2}}} = 1\]
By using the identity \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\] , here \[{a^m}\] is \[{5^x}\] and \[{2^{\left( {\dfrac{{3x - 3}}{x}} \right)}}\] , \[{a^n}\] is \[{5^3}\] and \[{2^2}\] .
So, we get \[ \Rightarrow {5^{x - 3}}*{2^{\left( {\dfrac{{x - 3}}{x}} \right)}} = 1\] .
Taking power \[x - 3\] common from the left hand side.
Therefore, we get \[{\left( {5*{2^{\dfrac{1}{x}}}} \right)^{\left( {x - 3} \right)}} = 1\] .
Taking log on both sides for simplifying the above equation and making the right hand side zero to solve and get the rational root.
As we know, log 1 is zero.
So, Now equation is \[\log \left( {{{\left( {5*{2^{\dfrac{1}{x}}}} \right)}^{\left( {x - 3} \right)}}} \right) = 0\]
Using property of log which is \[\log {n^m} = m*\log n\] . Here, n is \[5*{2^{\dfrac{1}{x}}}\] and m is \[\left( {x - 3} \right)\] .
So, after using this property, We get \[\left( {x - 3} \right)\log \left( {5*{2^{\dfrac{1}{x}}}} \right) = 0\]
Now we can separate the equation into two parts to find the rational roots of the equation.
\[ \Rightarrow \left( {x - 3} \right) = 0,\log \left( {5*{2^{\dfrac{1}{x}}}} \right) = 0\]
Therefore, \[{x_1} = 3,{\rm{ }}5*{2^{\dfrac{1}{x}}} = 1\]
Simplifying the second part by taking 5 on the right hand side which is equal to \[\dfrac{1}{5}\] .
\[{2^{\dfrac{1}{x}}} = \dfrac{1}{5}\]
Taking log on both sides and applying the property of log we applied earlier.
So, we get \[{x_2} = - {\log _5}2\]
Hence the roots of the \[{5^x}*\sqrt[x]{{{8^{x - 1}}}} = 500\] equation are \[{x_1} = 3\,{\rm{ and\, }}{x_2} = - {\log _5}2\] .

Note: To solve these types of questions, we need to firstly simplify the right hand side of the question. After simplification we can separate the equation to find the roots of the equation. If there is a power we can use property of log that is \[\log {n^m} = m*\log n\] for simplifying the equation.