Compare Expressions: 3² + √10² vs 2³ + √5·√20÷5

We want to calculate each of the expressions separately, however - in order to do this more efficiently, we will first deal with the multiplication terms between the roots in both expressions separately:

a. Let's start with the left expression, the multiplication of roots in this expression is:

$\sqrt{10}\cdot\sqrt{10}$

We'll apply the laws of exponents in order to simplify this expression, noting that the expression is actually multiplying the number by itself and therefore can be written as a term to the second power:

$\sqrt{10}\cdot\sqrt{10}=(\sqrt{10})^2$

Now let's recall the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

And the law of exponents for power to power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these two laws and calculate the value of the above expression:

$(\sqrt{10})^2 =(10^{\frac{1}{2}})^2=10^{\frac{1}{2}\cdot2}=10^1=10$

Where in the first step we converted the root in parentheses to a half power using the definition of root as a power mentioned earlier, and in the next step we applied the law of power to power that was also mentioned earlier, then we simplified the expression.

b. Let's continue to the multiplication of roots in the right expression:

$\sqrt{5}\cdot\sqrt{20}$

In addition to the definition of root as a power mentioned earlier, let's also recall the law of exponents for powers in parentheses where terms are multiplied but in the opposite direction:

$x^n\cdot y^n=(x\cdot y)^n$

The literal interpretation of this law in the direction given here is that a multiplication between two terms with equal power exponents can be written as a multiplication between the bases in parentheses raised to that same power,

Let's return to the expression in question and apply both laws of exponents mentioned:

$\sqrt{5}\cdot\sqrt{20} =5^{\frac{1}{2}}\cdot20^{\frac{1}{2}}=(5\cdot20)^{\frac{1}{2}}$

Where in the first step we converted the roots to half powers using the definition of root as a power, and in the next step we applied the last mentioned law of exponents in its specified direction, since both terms in the multiplication here have the same power,

Let's continue and simplify the expression we got:

$(5\cdot20)^{\frac{1}{2}} =100^{\frac{1}{2}}=\sqrt{100}=10$

Where in the first step we calculated the value of the multiplication in parentheses, in the next step we returned to writing roots using the definition of root as power, but in the opposite direction, in the final step we calculated the numerical value of the root,

Let's summarize a and b above, we got that:

$\sqrt{10}\cdot\sqrt{10}=(\sqrt{10})^2 =10$ and

$\sqrt{5}\cdot\sqrt{20} =(5\cdot20)^{\frac{1}{2}} =\sqrt{100}=10$,

Let's return to the original problem and use this information:

$3^2+\sqrt{10}\cdot\sqrt{10}\text{ }_{\textcolor{red}{—}\text{ }}2^3+\sqrt{5}\cdot\sqrt{20}:5 \\ \downarrow\\ 3^2+10\text{ }_{\textcolor{red}{—}\text{ }}2^3+10:5$

Let's continue and handle both expressions together, in the left expression we'll first calculate the value of the term in the power and then the result of the addition,

And in the right expression we'll first calculate the result of the term in the power and the result of the division operation and add between the results:

$3^2+10\text{ }_{\textcolor{red}{—}\text{ }}2^3+10:5 \\ \downarrow\\ 9+10\text{ }_{\textcolor{red}{—}\text{ }}8+2 \\ \downarrow\\ 19\text{ }_{\textcolor{red}{—}\text{ }}10$

Therefore the left expression gives a higher result, meaning the trend between the expressions is:

$19>10$

Therefore the correct answer is answer B.