Answer

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**Hint:**In this question, we will first take out the common factor \[\sqrt{5}\] from the numerator of the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\]. After that, we will do the rationalization. For rationalization, we will multiply the term \[\sqrt{2}\] in both denominator and numerator of the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] to solve this question easily. After that, we will put the values which are given in the question and then solve it, and then we will get the answer.

**Complete step by step solution:**

Let us solve this question.

In this question, we have given that the value of \[\sqrt{2}\] is 1.414, the value of \[\sqrt{3}\] is 1.732, the value of \[\sqrt{5}\] is 2.236, and the value of \[\sqrt{10}\] is 3.162

From these given values, we have to find the value of the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] up to three decimal places.

As we know that 10 can be written as 5 multiplied by 2 and 15 can be written as 3 multiplied by 5.

So, we can write the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] as

\[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{2\times 5}+\sqrt{3\times 5}}{\sqrt{2}}\]

The above equation can also be written as

\[\Rightarrow \dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{2}\times \sqrt{5}+\sqrt{3}\times \sqrt{5}}{\sqrt{2}}\]

Now, we can see that there is a common factor of \[\sqrt{5}\] in the numerator of the right side of the above equation.

So, after taking out that common factor, we can write the above equation as

\[\Rightarrow \dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{5}\left( \sqrt{2}+\sqrt{3} \right)}{\sqrt{2}}\]

Now, here we will do the rationalization. For that, we will multiply \[\sqrt{2}\] in both the numerator and denominator of the right side of the equation.

\[\Rightarrow \dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{2}\times \sqrt{5}\left( \sqrt{2}+\sqrt{3} \right)}{\sqrt{2}\times \sqrt{2}}\]

We can write the above equation as

\[\Rightarrow \dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{10}\left( \sqrt{2}+\sqrt{3} \right)}{2}\]

We know that \[\sqrt{2}=1.414\] and \[\sqrt{3}=1.732\], so \[\sqrt{2}+\sqrt{3}=1.414+1.732=3.146\]

The above equation can also be written as

\[\Rightarrow \dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{10}\left( 3.146 \right)}{2}=\sqrt{10}\times 1.573=3.162\times 1.573=4.973816\]

Hence, the value of \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] will be 4.973816

The value of \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] up to three places of decimal will be 4.974

This value is approx 4.975

**Hence, the option is B.**

**Note:**

We should know how to do the rationalization. We can solve this question by different methods.

We can write the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] after rationalizing, we get

\[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{2}\times \left( \sqrt{10}+\sqrt{15} \right)}{\sqrt{2}\times \sqrt{2}}=\dfrac{\sqrt{2}\times \sqrt{10}+\sqrt{2}\times \sqrt{15}}{2}\]

We can write the above term as

\[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}=\dfrac{\sqrt{2}\times \sqrt{10}+\sqrt{2}\times \sqrt{3}\times \sqrt{5}}{2}=\dfrac{1.414\times 3.162+1.414\times 1.732\times 2.236}{2}=\dfrac{9.947}{2}=4.9746\]

So, the value of the term \[\dfrac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}\] after rounding up to three places of decimal will be 4.975

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