Question

How do you simplify \[{{\left( \dfrac{56}{12} \right)}^{5}}\]?

Answer 1

Hint: In this problem, we have to simplify the given fraction. We know that, first we have to apply power to the terms, then we can divide the resulting number. We can first take the power to both numerator and the denominator and we can find the value of it, then we can divide the resulting value to get the simplified form. We can use a calculator to find the term with high power and the fraction after applying the power.

Complete step by step solution:
We know that the given fraction to be simplified is,
\[{{\left( \dfrac{56}{12} \right)}^{5}}\]
We can apply the whole power individually to the term in numerator and the denominator after simplifying it, we get
\[\Rightarrow {{\left( \dfrac{56}{12} \right)}^{5}}={{\left( \dfrac{14}{3} \right)}^{5}}\]
We can now find the value of the above numerator and the denominator.
\[\begin{align}
  & \Rightarrow {{\left( 14 \right)}^{5}}=14\times 14\times 14\times 14\times 14=537824 \\
 & \Rightarrow {{\left( 3 \right)}^{5}}=3\times 3\times 3\times 3\times 3=243 \\
\end{align}\]
Now we can substitute these values in the fraction, we get
\[\Rightarrow {{\left( \dfrac{56}{12} \right)}^{5}}={{\left( \dfrac{14}{3} \right)}^{5}}=\dfrac{{{\left( 14 \right)}^{5}}}{{{3}^{5}}}=\dfrac{537824}{243}\]
Now we can divide the above fraction using the calculator to get an approximate value.
\[\Rightarrow \dfrac{537824}{243}=2213\dfrac{65}{243}\]

Therefore, the simplified value of the given fractional expression \[{{\left( \dfrac{56}{12} \right)}^{5}}\] is \[2213\dfrac{65}{243}\].

Note: Students should know to apply the power to the individual terms in the numerator and the denominator from the whole power. We can analyse that the terms in the numerator and the denominator are with high power and the fraction after applying the power is in a complicated form. In that case we can use a calculator to find the term with high power and the fraction after applying the power. We will get a simplified form.