# The simplified value of $\sqrt{72}+\sqrt{800}-\sqrt{18}$ is :

Last updated date: 19th Mar 2023

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Hint: The given problem is related to the square root of numbers. Find the square root of the numbers by factorization and then use mathematical operations to evaluate the simplified value of the given expression.

Complete step-by-step answer:

We are asked to find the simplified value of $\sqrt{72}+\sqrt{800}-\sqrt{18}$. First, we will evaluate the value of each term, then find the simplified value of the expression. To find the value of each term, we will determine the value of the square root by factorization.

We know, $72=2\times 2\times 2\times 3\times 3$. So, $\sqrt{72}=\sqrt{2\times 2\times 2\times 3\times 3}$ . We will express the factors as a product of squares of prime numbers. So, $\sqrt{72}=\sqrt{{{2}^{2}}\times {{3}^{2}}\times 2}$.

$=2\times 3\times \sqrt{2}$

$=6\sqrt{2}$

Now, $800=2\times 2\times 2\times 2\times 2\times 5\times 5$ . So, $\sqrt{800}=\sqrt{2\times 2\times 2\times 2\times 2\times 5\times 5}$ . We will express the factors as a product of squares of prime numbers. So, \[\sqrt{800}=\sqrt{{{2}^{2}}\times {{2}^{2}}\times {{5}^{2}}\times 2}\] .

\[=2\times 2\times 5\times \sqrt{2}\]

\[=20\sqrt{2}\]

Now, $18=2\times 3\times 3$ . So, $\sqrt{18}=\sqrt{2\times 3\times 3}$ . We will express the factors as a product of squares of prime numbers. So, $\sqrt{18}=\sqrt{{{3}^{2}}\times 2}$ .

$=3\sqrt{2}$

Now, we have the values of the square root of all the terms given in the expression. Now, we can find the value of the expression. The given expression is $\sqrt{72}+\sqrt{800}-\sqrt{18}$ . We have calculated the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ as $20\sqrt{2}$ , $3\sqrt{2}$ , and $6\sqrt{2}$ respectively. Now, we will substitute the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ in the given expression. On substituting the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ in the given expression, we get:

$\sqrt{72}+\sqrt{800}-\sqrt{18}=6\sqrt{2}+20\sqrt{2}-3\sqrt{2}$

Now, we will take $\sqrt{2}$ common from all three terms. On taking $\sqrt{2}$ common from all three terms, we get $\sqrt{72}+\sqrt{800}-\sqrt{18}=\left( 6+20-3 \right)\sqrt{2}=23\sqrt{2}$.

Hence, the simplified value of the expression $\sqrt{72}+\sqrt{800}-\sqrt{18}$ is equal to $23\sqrt{2}$ .

Note: While evaluating the square root of a number, it is better to express the number as a product of its prime factors. This way, it will be easier to calculate the square root and there will be no confusion while evaluating the square root.

Complete step-by-step answer:

We are asked to find the simplified value of $\sqrt{72}+\sqrt{800}-\sqrt{18}$. First, we will evaluate the value of each term, then find the simplified value of the expression. To find the value of each term, we will determine the value of the square root by factorization.

We know, $72=2\times 2\times 2\times 3\times 3$. So, $\sqrt{72}=\sqrt{2\times 2\times 2\times 3\times 3}$ . We will express the factors as a product of squares of prime numbers. So, $\sqrt{72}=\sqrt{{{2}^{2}}\times {{3}^{2}}\times 2}$.

$=2\times 3\times \sqrt{2}$

$=6\sqrt{2}$

Now, $800=2\times 2\times 2\times 2\times 2\times 5\times 5$ . So, $\sqrt{800}=\sqrt{2\times 2\times 2\times 2\times 2\times 5\times 5}$ . We will express the factors as a product of squares of prime numbers. So, \[\sqrt{800}=\sqrt{{{2}^{2}}\times {{2}^{2}}\times {{5}^{2}}\times 2}\] .

\[=2\times 2\times 5\times \sqrt{2}\]

\[=20\sqrt{2}\]

Now, $18=2\times 3\times 3$ . So, $\sqrt{18}=\sqrt{2\times 3\times 3}$ . We will express the factors as a product of squares of prime numbers. So, $\sqrt{18}=\sqrt{{{3}^{2}}\times 2}$ .

$=3\sqrt{2}$

Now, we have the values of the square root of all the terms given in the expression. Now, we can find the value of the expression. The given expression is $\sqrt{72}+\sqrt{800}-\sqrt{18}$ . We have calculated the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ as $20\sqrt{2}$ , $3\sqrt{2}$ , and $6\sqrt{2}$ respectively. Now, we will substitute the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ in the given expression. On substituting the values of $\sqrt{72},\sqrt{800}$ and $\sqrt{18}$ in the given expression, we get:

$\sqrt{72}+\sqrt{800}-\sqrt{18}=6\sqrt{2}+20\sqrt{2}-3\sqrt{2}$

Now, we will take $\sqrt{2}$ common from all three terms. On taking $\sqrt{2}$ common from all three terms, we get $\sqrt{72}+\sqrt{800}-\sqrt{18}=\left( 6+20-3 \right)\sqrt{2}=23\sqrt{2}$.

Hence, the simplified value of the expression $\sqrt{72}+\sqrt{800}-\sqrt{18}$ is equal to $23\sqrt{2}$ .

Note: While evaluating the square root of a number, it is better to express the number as a product of its prime factors. This way, it will be easier to calculate the square root and there will be no confusion while evaluating the square root.

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