Answer
$0.5187$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the Laws of Logarithms to find an equivalent expression for the given expression, $
\log_{10} 36^{1/3}
,$ which uses the given logarithmic values, $\log_{10} 2=
0.3010
$ and/or $\log_{10} 3=
0.4771
.$
$\bf{\text{Solution Details:}}$
The given expression can be expressed as
\begin{array}{l}\require{cancel}
\log_{10} (4\cdot9)^{1/3}
\\\\=
\log_{10} (2^2\cdot3^2)^{1/3}
.\end{array}
Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\log_{10} 2^{2\cdot\frac{1}{3}}\cdot3^{2\cdot\frac{1}{3}}
\\\\=
\log_{10} 2^{\frac{2}{3}}\cdot3^{\frac{2}{3}}
.\end{array}
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the given expression is equivalent
\begin{array}{l}\require{cancel}
\log_{10} 2^{\frac{2}{3}}+\log_{10}3^{\frac{2}{3}}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{2}{3}\log_{10} 2+\dfrac{2}{3}\log_{10}3
\\\\=
\dfrac{2}{3}(\log_{10} 2+\log_{10}3)
.\end{array}
Substituting the known values of the logarithmic expressions results to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}(0.3010+0.4771)
\\\\=
\dfrac{2}{3}(0.7781)
\\\\=
0.5187
.\end{array}
