Question

How do you use the law of exponents to simplify the expression $ {\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} $

Answer 1

Hint: As you can see in the question the bases are the same but the powers are different which indicate to us that we have to use the laws of indices. But the expression will not be simplified using any one law of indices. You will have to use various different laws of indices as required. Start the process within the bracket first.

Complete step-by-step answer:
The expression that is given above looks scary but can be broken down if proper laws are applied.
Looking at the question first thing you can spot is that the bases are the same and their powers are differing which is a good sign as we know that when the bases are same and the powers differ it’s time to use the laws of indices.
Since there are multiple indices present first we will try to sort things within the bracket.
We are going to use $ \dfrac{1}{{{a^n}}} = {a^{ - n}} $
Hence the above expression will become
 $ {\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} $ = $ {\left( {{3^2}{{.3}^3}} \right)^{\dfrac{3}{5}}} $
Now using $ {a^m}.{a^n} = {a^{m + n}} $ we get
\[{\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} = {\left( {{3^{2 + 3}}} \right)^{\dfrac{3}{5}}}\]
 $ \Rightarrow {\left( {{3^5}} \right)^{\dfrac{3}{5}}} $
Now using $ {\left( {{a^n}} \right)^m} = {a^{n \times m}} $ , we get
\[{\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} = {3^{5 \times \dfrac{3}{5}}}\]
which solving further we get
\[{\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} = {3^3} = 27\]
hence this is the final value of the above equation.\[{\left( {\dfrac{{{3^2}}}{{{3^{ - 3}}}}} \right)^{\dfrac{3}{5}}} = {3^3} = 27\]
So, the correct answer is “27”.

Note: State the law first and then apply it. Identification of laws to be used in this kind of expression is utmost necessary. As this will decide whether the expression will get simplified or not. Also the laws should be applied in order while solving the expression as this can lead to many mathematical errors.