Question

How do you find the value of ${\left( {\dfrac{{216}}{{1000}}} \right)^{\dfrac{2}{3}}}$ ?

Answer 1

Hint: Firstly, write both the constants in the numerator and denominator of the bigger fraction in exponential form. They must be written in powers of smaller constants such as $2,3,5...\;$ . Now multiply the powers with the smaller fractions which are acting as a power to the bigger fraction and then simplify and then evaluate to get a constant or fraction as a final solution.

Complete step-by-step answer:
The given fraction is, ${\left( {\dfrac{{216}}{{1000}}} \right)^{\dfrac{2}{3}}}$
Now write the numerator of the bigger fraction in exponent form.
$\Rightarrow 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
Now we can raise to powers as,
$\Rightarrow 216 = {2^3} \times {3^3}$
Now do the same to the denominator of the bigger fraction.
$\Rightarrow 1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5$
Can be also written as,
$\Rightarrow 1000 = {2^3} \times {5^3}$
Now write it all together.
$\Rightarrow {\left( {\dfrac{{{{(2 \times 3)}^3}}}{{{{(2 \times 5)}^3}}}} \right)^{\dfrac{2}{3}}}$
Now bring the inner powers to the outside and multiply them.
$\Rightarrow {\left( {\dfrac{{(2 \times 3)}}{{(2 \times 5)}}} \right)^{3 \times \dfrac{2}{3}}}$
Now evaluate the outside fraction which is the power to the bigger fraction.
$\Rightarrow {\left( {\dfrac{{(2 \times 3)}}{{(2 \times 5)}}} \right)^2}$
Now there are also some common terms in the numerator and denominator, we can cancel them.
$\Rightarrow {\left( {\dfrac{3}{5}} \right)^2}$
Now we expand back from the exponential form.
We get,
$\Rightarrow \dfrac{9}{{25}}$

$\therefore {\left( {\dfrac{{216}}{{1000}}} \right)^{\dfrac{2}{3}}} = \dfrac{9}{{25}}$

Additional information: If an improper fraction is given, we must first convert it to a mixed fraction. For that, we must first divide the denominator with the numerator. Then we write the answer in the whole number part i.e., in front of the fraction. Now, we write the remainder in the numerator part and the denominator remains as it is. A mixed fraction is a whole number and a fraction part combined. The units will always remain the same.

Note:
Always write the multiplied product in expanded form to easily cancel out the terms in the numerator and the denominator. Then evaluate to get the final answer which might be a fraction or a constant.