Question

How do you evaluate ${(0.2)^4}?$

Answer 1

Hint:To evaluate ${(0.2)^4}$, first convert the decimal number into fractional form in order to make the process easy. After converting into fractional form, evaluate the exponent and then finally convert the fraction into decimal.

Completed step by step solution:
In order to evaluate ${(0.2)^4}$, we will convert the decimal number
$0.2$ into fractional form,
Since $0.2$ has only one digit after the decimal point, so multiplying and dividing it by ${10^1} = 10$ in
order to get the fractional form of $0.2$
$
= 0.2 \times \dfrac{{10}}{{10}} \\
= \dfrac{2}{{10}} \\
$
So we get the fractional of decimal number $0.2 = \dfrac{2}{{10}}$
Now evaluating \[{(0.2)^4}\;as\;{\left( {\dfrac{2}{{10}}} \right)^4}\]
$
= {(0.2)^4} \\
= {\left( {\dfrac{2}{{10}}} \right)^4} \\
= \dfrac{{{2^4}}}{{{{10}^4}}} \\
$
Calculating the value of ${2^4}\;{\text{and}}\;{10^4}$
$
\Rightarrow {2^4} = 2 \times 2 \times 2 \times 2\;{\text{and}}\;{10^4} = 10 \times 10 \times 10
\times 10 \\
\Rightarrow {2^4} = 16\;{\text{and}}\;{10^4} = 10000 \\
$
Putting values of ${2^4}\;{\text{and}}\;{10^4}$ in the above fraction in order to evaluate further
$
= \dfrac{{{2^4}}}{{{{10}^4}}} \\
= \dfrac{{16}}{{10000}} \\
$
Now converting the resulting fraction into decimal to get the required answer in decimals
$ = \dfrac{{16}}{{10000}}$
Here in the fraction, the denominator is in the power of $10$ , we can also write the resultant fraction as $ = \dfrac{{16}}{{{{10}^4}}}$
So putting the decimal four digits before the right, since $16$ is a two digit number, we will put zeros before it in order to put the decimal point.
$
= \dfrac{{16}}{{{{10}^4}}} \\
= 0.0016 \\
$
Therefore $0.0016$ is the required answer.

Note: We have converted the decimal number into fraction to make the calculation easy and understandable either you can also evaluate this question by directly multiplying the given number to itself up to the exponent times and put the decimal point the n digits before from the right, where n is the product of the number of digits after the decimal point of given number and the exponent. But this process is lengthy one if the exponent is a larger number.