Question

How will you evaluate and simplify ${\left( {27} \right)^{\dfrac{2}{3}}}$ ?

Answer 1

Hint:
In this question, we have been given with the fractional exponent. To find the value of ${\left( {27} \right)^{\dfrac{2}{3}}}$, we will find the 3rd root of 27 and then raise that value to the power of 2. By following this way we can find the answer.

Complete step by step solution:
Rule of fractional exponents states that if any number ‘\[a\]’ raised to the power ‘’, then it can be evaluated as \[{a^{\dfrac{b}{c}}} = {\left( {\sqrt[c]{a}} \right)^b}\]
If we compare the above equation with the question, we get,
\[
  a = 27 \\
  b = 2 \\
  c = 3 \\
 \]
The equation can be written as,
\[{27^{\dfrac{2}{3}}} = {\left( {\sqrt[3]{{27}}} \right)^2}\]
Firstly, we will find the \[{3^{rd}}\] root of \[27\]
\[\sqrt[3]{{27}} = 3\]
Since, \[{3^3} = 27\]
Now, we will raise the value i.e. \[{3^{rd}}\] root of \[27\]to the power of \[2\], we get,
\[
  {\left( {\sqrt[3]{{27}}} \right)^2} = {\left( 3 \right)^2} = 9 \\
   \Rightarrow {27^{\dfrac{2}{3}}} = 9 \\
 \]

Additional information:
A fractional exponent is a term which represents the powers and roots together. Exponents are nothing but the power or indices. The number under the radical \[\sqrt {} \] is called a radicand. The root being taken is referred to as the index or order. And the power tells us about how many times we need to multiply the value of root obtained

Note:
Students should use the sign conventions properly while solving the problem. Basic roots of numbers should be known by the students. Tables should be memorized by the students to ease the difficulties and problems faced by students while understanding and solving such problems. Students should memorize the formula for fractional exponents. Students should be able to identify the proper substituent i.e. \[a,b,c\] and put it in the formula.