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# Simplify:- $\sqrt {72} + \sqrt {800} - \sqrt {18}$

Last updated date: 12th Aug 2024
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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, $\sqrt {72} + \sqrt {800} - \sqrt {18} = 23\sqrt 2$