Question

How do you write in a simplified radical form of \[4\sqrt {48} ?\]

Answer 1

Hint:The given question involves the operation of addition/ subtraction/ multiplication/ division. To solve this problem we need to know the square value of basic terms. We need to know the meaning of radical which is given in the question. Also, we need to know the relation between the square function and square root function to make an easy calculation.

Complete step by step solution:
The given question is shown below,
\[4\sqrt {48} = ? \to \left( 1 \right)\]
Here radical means, the final answer wouldn’t have any more square roots of cubic roots.
We know that \[48\] can also be written as,
\[48 = 6 \times 8\]
So, the equation\[\left( 1 \right)\] becomes,
\[\left( 1 \right) \to 4\sqrt {48} = ?\]
\[4\sqrt {48} = 4\sqrt {6 \times 8} \to \left( 2 \right)\]
We know that \[8\] can also be written as \[2 \times 2 \times 2\]and\[6\] can also be written as \[2 \times 3\]. So, the equation \[\left( 2 \right)\] becomes
\[\left( 2 \right) \to 4\sqrt {48} = 4\sqrt {6 \times 8} \]
\[4\sqrt {48} = 4\sqrt {6 \times 8} = 4\sqrt {2 \times 3 \times 2 \times 2 \times 2} \]
The above equation can also be written as,
\[4\sqrt {48} = 4\sqrt {{2^2} \times {2^2} \times 3} \]
So, we know that \[\sqrt {{2^2}} = 2\]. So, we get
\[4\sqrt {48} = 4 \times 2 \times 2 \times \sqrt 3 \]
\[4\sqrt {48} = 16\sqrt 3 \]
\[\sqrt 3 \] cannot be simplified further. So, it is in the form of radical.
So, the final answer is,
The radical form of \[4\sqrt {48} = 16\sqrt 3 \].

Note: This question describes the arithmetic operations of addition/ subtraction/ multiplication/ division. Note that for these types of questions the final answer would be in radical form. This means the final answer wouldn’t have any more square roots of cubic roots. Note that \[\sqrt {{n^2}} \] is equal to the value of \[n\]. When multiplying different sign numbers we would remember the following things,
1) When a positive number is multiplied with a positive number the final answer becomes a
positive number.
2) When a negative number is multiplied with the positive number the answer becomes a
negative number.
3) When a negative number is multiplied with a negative number the answer becomes a
positive number.