Question

question
find the sum of $5\sqrt{3}$ and $4\sqrt{10}$ in simplest form. also, determine whether the result is rational or irrational and explain your answer.
answer attempt 1 out of 3
result:
the result is because it be written as the ratio of two integers and its decimal expansion terminate or repeat.

Explanation:

Step1: Check for like radicals

Radicals can be added only if they have the same radicand (the number inside the square root). Here, the radicands are \( 3 \) and \( 10 \), which are different. So, \( 5\sqrt{3} \) and \( 4\sqrt{10} \) cannot be combined into a single radical term. Thus, the sum is \( 5\sqrt{3} + 4\sqrt{10} \).

Step2: Determine if the result is rational

A rational number can be written as \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b
eq 0 \). \( \sqrt{3} \) and \( \sqrt{10} \) are irrational numbers (their decimal expansions are non - terminating and non - repeating). The sum of a non - zero irrational number (\( 5\sqrt{3} \)) and another non - zero irrational number (\( 4\sqrt{10} \)) is irrational. An irrational number cannot be written as the ratio of two integers, and its decimal expansion does not terminate or repeat.

Result (Sum):

\( 5\sqrt{3}+4\sqrt{10} \)

The result is:

Irrational, because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.

Answer:

The sum is \( \boldsymbol{5\sqrt{3} + 4\sqrt{10}} \). The result is irrational because it cannot be written as the ratio of two integers and its decimal expansion does not terminate or repeat.