find the sum of 5 radical 3 and 4 radical 10 in simplest form. Also, determine whether the result is rational or irrational and explain your answer.

To find the sum of \( 5\sqrt{3} \) and \( 4\sqrt{10} \), we start by adding the two terms:

\[ 5\sqrt{3} + 4\sqrt{10} \]

Since these terms involve different radical expressions (\(\sqrt{3}\) and \(\sqrt{10}\)), they cannot be combined further in terms of simplification. Therefore, the result can be expressed as:

\[ 5\sqrt{3} + 4\sqrt{10} \]

Next, we analyze whether this result is rational or irrational. A number is rational if it can be expressed as the ratio of two integers. In contrast, it is irrational if it cannot be expressed in that way.

Both \( \sqrt{3} \) and \( \sqrt{10} \) are irrational numbers. When we multiply irrational numbers by rational coefficients (in this case, the rational numbers 5 and 4), the results \( 5\sqrt{3} \) and \( 4\sqrt{10} \) remain irrational. The sum of two irrational numbers is not always irrational, but in this case, since \( 5\sqrt{3} \) and \( 4\sqrt{10} \) do not have a common radical part, their sum is indeed irrational.

Therefore, the final statement about the result should be completed as follows:

\[ \text{result: } 5\sqrt{3} + 4\sqrt{10} \]

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

A. \( 5\sqrt{3} + 4\sqrt{10} \)
B. \( \text{can or cannot} \) - cannot
C. \( \text{does or does not} \) - does not

So, the final answer is:

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

The result \( 5\sqrt{3} + 4\sqrt{10} \) is irrational.

Explanation:

1. Irrationality of the Terms:

   • Both terms \( \sqrt{3} \) and \( \sqrt{10} \) are irrational numbers. This means they cannot be expressed as the ratio of two integers.

2. Sum of Irrational Numbers:

   • The products \( 5\sqrt{3} \) and \( 4\sqrt{10} \) are also irrational because multiplying a nonzero rational number (5 or 4) by an irrational number results in an irrational number.

3. Adding Different Irrationals:

   • The sum of two irrational numbers is not necessarily irrational, but in this case (\( 5\sqrt{3} \) and \( 4\sqrt{10} \)), they do not simplify to a common form and thus their sum remains irrational.

Conclusion:

Therefore, \( 5\sqrt{3} + 4\sqrt{10} \) is irrational.