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Simplify:- \[\sqrt {72} + \sqrt {800} - \sqrt {18} \]

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Last updated date: 22nd Jun 2024
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Answer
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Hint:
Here, we have to simplify the expression. We will simplify the surds to the whole integers by using the BODMAS rule and Rules of surds. Simplification is the process of sorting out all the arithmetic operations and solving the expression to find an integer.

Formula Used:
Rule of surds: \[a\sqrt c \pm b\sqrt c = \left( {a \pm b} \right)\sqrt c \]

Complete Step by step Solution:
We are given with a mathematical expression \[\sqrt {72} + \sqrt {800} - \sqrt {18} \]
We will find the factors of the given numbers to simplify the expression in the form of surds.
First, we will do the factorization of 72.
\[\begin{array}{l}{\rm{2}}\left| \!{\underline {\,
  {{\rm{72}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{36}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{18}}} \,}} \right. \\{\rm{3}}\left| \!{\underline {\,
  {\rm{9}} \,}} \right. \\{\rm{3}}\left| \!{\underline {\,
  {\rm{3}} \,}} \right. \end{array}\]
We can write 72 as \[72 = 2 \times 2 \times 2 \times 3 \times 3\]
So, that the number can be written as \[\sqrt {72} = \sqrt {2 \times 2 \times 2 \times 3 \times 3} \]
Now, we will do the factorization of 800.
\[\begin{array}{l}{\rm{2}}\left| \!{\underline {\,
  {{\rm{800}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{400}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{200}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{100}}} \,}} \right. \\{\rm{2}}\left| \!{\underline {\,
  {{\rm{50}}} \,}} \right. \\{\rm{5}}\left| \!{\underline {\,
  {{\rm{25}}} \,}} \right. \\{\rm{5}}\left| \!{\underline {\,
  {\rm{5}} \,}} \right. \end{array}\]
We can write 800 as \[800 = 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5\]
So, that the number can be written as \[\sqrt {800} = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5} \]
Now, we will do the factorization of 18.
\[\begin{array}{l}{\rm{2}}\left| \!{\underline {\,
  {{\rm{18}}} \,}} \right. \\{\rm{3}}\left| \!{\underline {\,
  {\rm{9}} \,}} \right. \\{\rm{3}}\left| \!{\underline {\,
  {\rm{3}} \,}} \right. \end{array}\]
We can write 800 as
So, that the number can be written as \[\sqrt {18} = \sqrt {2 \times 3 \times 3} \]
Now, simplifying the expression, we get
So, the factors can be paired and by taking square root on both the sides, we get
\[\sqrt {72} = 6\sqrt 2 \]
\[\sqrt {800} = 20\sqrt 2 \]
\[\sqrt {18} = 3\sqrt 2 \]
Now substituting the values in the given expression, we get
\[\sqrt {72} + \sqrt {800} - \sqrt {18} = 6\sqrt 2 + 20\sqrt 2 - 3\sqrt 2 \]
Now, by using the BODMAS Rule and by using the rules of surds, we get
\[ \Rightarrow \sqrt {72} + \sqrt {800} - \sqrt {18} = 26\sqrt 2 - 3\sqrt 2 \]
Now, subtracting the surds using the rules of surds, we get
\[ \Rightarrow \sqrt {72} + \sqrt {800} - \sqrt {18} = 23\sqrt 2 \]

Therefore, \[\sqrt {72} + \sqrt {800} - \sqrt {18} = 23\sqrt 2 \]

Additional Information:
BODMAS rule states that the first operation has to be done which is in the brackets, next the operation applies on the indices or order, then it moves on to the division and multiplication and then using addition and subtraction we will simplify the expression. If addition or subtraction and division or multiplication are in the same calculations, then it has to be done from left to right.

Note:
We know that surds are the numbers which are not the perfect squares. Surds cannot be expressed as a rational number or whole number. A surd has only one term called simple surds. An expression that has the same surds can be added, subtracted, multiplied or divided. Thus the surds are compound surds. Surds that are completely irrational are called pure surds.