Question

How do you simplify \[4\] square root \[8 - 3\] square root \[2\]?

Answer 1

Hint: In order to solve this question first, we assume a variable equal to the expression and then convert this equation in mathematical format, and then we will make the same irrational number with integer multiplication and then we simplify that relation in the form of an integer with a multiplication of irrational number.

Complete answer:
First we convert this equation into a mathematical expression.
\[4\] square root \[8 - 3\] square root \[2\] is written in mathematical expression as \[4\sqrt 8 - 3\sqrt 2 \]
Now we assume a variable for the mathematical expression.
\[x = 4\sqrt 8 - 3\sqrt 2 \]
We know that \[\sqrt 2 \] is an irrational number.
So we are converting \[\sqrt 8 \] to \[\sqrt 2 \] in the equation.
\[x = 4\sqrt {4 \times 2} - 3\sqrt 2 \]
On separating the terms from the first square root.
\[x = 4\sqrt 4 \sqrt 2 - 3\sqrt 2 \]
We know the value of \[\sqrt 4 \] is \[2\]
On putting this value in the last equation.
\[x = 4 \times 2\sqrt 2 - 3\sqrt 2 \]
On further solving
\[x = 8\sqrt 2 - 3\sqrt 2 \]
Now on taking the irrational part common
\[x = \left( {8 - 3} \right)\sqrt 2 \]
On further solving
\[ \Rightarrow x = 5\sqrt 2 \]
Final answer:
The final value of the given expression \[4\] square root \[8 - 3\] square root \[2\] is \[5\sqrt 2 \].

Note:
In solving these types of questions students are not able to find what they have to do or they are directly finding the value of the irrational parts and put in the expression and then multiply and find the answer in decimal format. Students have to remember that we have to separate the irrational part and rational part as a multiple of integer and also keep the final answer in the same format. If two different irrational parts are given then you get a longer expression and try to take common irrational terms if possible and then keep both irrational numbers separate.