Question

How do you simplify ${{\left( 2{{a}^{5}} \right)}^{3}}$?

Answer 1

Hint: To solve this question, we need to use the properties of the exponents. A product of two terms raised to a power can be written as the product of each term raised to the same power, that is, ${{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}$. Also, when a term raised to a power is itself raised to some power, then it can be written as the term raised to the multiplication of two powers, that is, ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$. Applying these two properties on the given expression will simplify it, and the simplified expression will be the final answer.

Complete step by step answer:
Let us represent the term given to us in the question as
$E={{\left( 2{{a}^{5}} \right)}^{3}}$
Now, from the properties of the exponents we know that ${{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}$. Applying this in the above expression, we can rewrite the given expression as
$\Rightarrow E={{\left( 2 \right)}^{3}}{{\left( {{a}^{5}} \right)}^{3}}$
We know that ${{\left( 2 \right)}^{3}}=2\times 2\times 2=8$. Putting this in the above expression, we get
$\Rightarrow E=8{{\left( {{a}^{5}} \right)}^{3}}$
Now, from the other property of the exponent, we also know that ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{xy}}$. Applying this in the above expression, we finally get the simplified expression as
\[\begin{align}
  & \Rightarrow E=8{{a}^{5\times 3}} \\
 & \Rightarrow E=8{{a}^{15}} \\
\end{align}\]
Hence, the simplified expression of the given expression ${{\left( 2{{a}^{5}} \right)}^{3}}$ is \[8{{a}^{15}}\].

Note:
If we do not remember the properties of the exponents or the powers, then we can expand the given expression as a product of the terms according to the value of the power. Then carrying out the necessary multiplication and applying the addition rule of exponents, we will get the final simplified expression. But we must note that we have to remember at least the addition rule of exponents, otherwise we will not be able to simplify the given expression.