Question

What is $(2 times\ square\ root\ of\ 2)^3$?

Answer 1

Hint: These types of questions are based on simplifying the expressions. Through the word problem, we had to sort it out and had to write it in an expression form and solve it further. In order to solve a type of expression we will get through our asked question that is we had to know how to solve the root and square and cubic terms.

Complete step by step solution:
Some identities which would help to solve the question are:
$\begin{align}
  & \sqrt{a}\times \sqrt{a}=a \\
 & {{a}^{n}}=a\times a\times a\times ........\text{n times} \\
\end{align}$
To make a word problem into an expression, we will start by reading question carefully, that is
2(times square root of 2)^3 where times means multiplication.
^x means it is exponentially on the value.
So we can say that the square root of 2 can be written as
$\sqrt{2}$
Now, we have been given that 2 times the square root of 2, so we can write it as
$2\times \sqrt{2}$
And then we have (2 times square root of 2)^3 which further can be written as
${{\left( 2\times \sqrt{2} \right)}^{3}}$
Now solving it further, using the identities of powers and exponents, we will get
\[\begin{align}
  & {{\left( 2\times \sqrt{2} \right)}^{3}} \\
 & = {{2}^{3}}\times {{\left( \sqrt{2} \right)}^{3}} \\
\end{align}\]
Now, after opening brackets, we will get
\[= 8\times \left( \sqrt{2}\times \sqrt{2}\times \sqrt{2} \right)\]
And it can be further expressed as

\[\begin{align}
  & = 8\times \left( 2\times \sqrt{2} \right) \\
 & = 8\times 2\times \sqrt{2} \\
 & = 16\times \sqrt{2} \\
 & = 16\sqrt{2} \\
\end{align}\]
As we have used the above identities to solve the expression further.
Hence the answer is \[16\sqrt{2}\].

Note: To solve the expression you must know the basic exponent algebraic identities, which are mentioned above. Moreover, avoid direct solving these types of problems, try to make an expression and solve it further.