Question

What is $\dfrac{1}{3}$ to the 5th power?

Answer 1

Hint: To find the 5th power of $\dfrac{1}{3}$, we are going to use the method of log. First of all, let the fifth power of $\dfrac{1}{3}$ be x. Now, the fifth power of $\dfrac{1}{3}$ can also be denoted by raising $\dfrac{1}{3}$ to 5. So, we will get the equation as $x = {\left( {\dfrac{1}{3}} \right)^5}$.Now, take log on both sides and then simplify RHS. After that, to find the value of x, take antilog on both sides and we will get our answer.

Complete step-by-step solution:
In this question, we are supposed to find the 5th power of $\dfrac{1}{3}$.
That means we have to find $\dfrac{1}{3}$ raised to 5.
$ \to {\left( {\dfrac{1}{3}} \right)^5}$
We can easily find this using a calculator, but we are going to see a method to find this without using a calculator.
For this, we are going to use the log method.
Let the 5th power of $\dfrac{1}{3}$ be $x$.
$ \Rightarrow x = {\left( {\dfrac{1}{3}} \right)^5}$- - - - - - - - - - - (1)
Now, to find the 5th power of $\dfrac{1}{3}$ using the log method, introduce log on both sides of the equation.
Therefore, equation (1) becomes
$ \Rightarrow \log x = \log {\left( {\dfrac{1}{3}} \right)^5}$ - - - - - - - - (2)
Now, we have the property $\log {a^b} = b\log a$. Therefore, equation (2) becomes
$ \Rightarrow \log x = 5\log \left( {\dfrac{1}{3}} \right)$- - - - - - - (3)
Now, using the property $\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$, equation (3) becomes
$ \Rightarrow \log x = 5\left( {\log 1 - \log 3} \right)$- - - - - - - - (4)
Now, the value of $\log 1 = 0$ and $\log 3 = 0.477121$. Therefore, equation (4) becomes
$ \Rightarrow \log x = - 5\left( {0.477121} \right)$
$ \Rightarrow \log x = - 2.385606 $
Now, we need the value of x. So, take antilog on both sides, we get
$\Rightarrow x = anti\log \left( { - 2.38560628} \right) \\
   \Rightarrow x = {\text{0}}{\text{.004115226}} \\ $
Hence, the fifth power of $\dfrac{1}{3}$ is \[{\text{0}}{\text{.004115226}}\].

Note: We can also solve this question with another method.
${\left( {\dfrac{1}{3}} \right)^5} = \dfrac{{1 \times 1 \times 1 \times 1 \times 1}}{{3 \times 3 \times 3 \times 3 \times 3}}$
Multiply 3 5 times in denominator.
 ${\left( {\dfrac{1}{3}} \right)^5} = \dfrac{{1 \times 1 \times 1 \times 1 \times 1}}{{3 \times 3 \times 3 \times 3 \times 3}} = \dfrac{1}{{243}}$
Now, divide 1 by 243.
${\left( {\dfrac{1}{3}} \right)^5} = \dfrac{{1 \times 1 \times 1 \times 1 \times 1}}{{3 \times 3 \times 3 \times 3 \times 3}} = \dfrac{1}{{243}} = 0.004115226$