Question

Simplify $\log \sqrt[4]{{729\sqrt[3]{{{9^{ - 1}}{{.27}^{ - \dfrac{4}{3}}}}}}}$

Answer 1

Hint : Here in this question we have to find or determine the value of the logarithmic function, but the term is in the form of the cubic root and fourth root. So first we simplify the root terms and then we obtain the value for that and then we apply the log to the obtained answer hence by this way we can simplify the given question.

Complete step-by-step answer :
The logarithmic function is called the inverse of the exponential function. The logarithmic function has two kinds namely, natural logarithmic function and common logarithmic function.
Now here the log function’s exponent is in the form of the roots so first we simplify the roots and then we determine the log for that value.
Now consider the question
$= \log \sqrt[4]{{729\sqrt[3]{{{9^{ - 1}}{{.27}^{ - \dfrac{4}{3}}}}}}}$
First we will simplify the ${9^{ - 1}}{.27^{ - \dfrac{4}{3}}}$, so on simplifying we have
$= {9^{ - 1}}{.27^{ - \dfrac{4}{3}}}$
The power for the number is negative so it can be written in the form of fraction
$= \dfrac{1}{{{9^1}}}.\dfrac{1}{{{{27}^{\dfrac{4}{3}}}}}$
on simplifying this we have
$= \dfrac{1}{9}.\dfrac{1}{{{{\left( {{{\left( {27} \right)}^4}} \right)}^{\dfrac{1}{3}}}}}$
We can interchange the power also
$= \dfrac{1}{9}.\dfrac{1}{{{{\left( {{{\left( {27} \right)}^{\dfrac{1}{3}}}} \right)}^4}}}$
The cube root of 27 is 3, so now we have
$= \dfrac{1}{9}.\dfrac{1}{{{{\left( 3 \right)}^4}}}$
On further simplification we have
$= \dfrac{1}{9}.\dfrac{1}{{81}}$
On multiplying these fractions we have
$= \dfrac{1}{{729}}$
The given question can be written as
$= \log \sqrt[4]{{729\sqrt[3]{{\dfrac{1}{{729}}}}}}$
On applying the cube root for $\dfrac{1}{{729}}$ we get
$= \log \sqrt[4]{{729.\dfrac{1}{9}}}$
On dividing 729 by 9 we get
$= \log \sqrt[4]{{81}}$
The number 81 can be written as
$= \log \sqrt[4]{{3 \times 3 \times 3 \times 3}}$
The number 3 is multiplied 4 times, it can be written in the terms of exponent and it is
$= \log \sqrt[4]{{{3^4}}}$
The fourth power and forth root will gets cancels so now we have
$= \log 3$
The value of $\log 3 = 0.4771$, by using the scientific calculator we get the answer.
Hence we have simplified the given question.
So, the correct answer is “$\log 3 = 0.4771$”.

Note : When the number is multiplied twice by the number itself then we can say it has a square, if it is multiplied thrice we call it a cube and so on. The square and square root, cube and cube root, forth power and fourth root are inverse of one another. To find the root of a given number we use the division method. Hence we can simplify the number with the help of a table of multiplication.