Question
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.
result: ?
the result is blank, because it blank, be written as the ratio of two integers and its decimal expansion blank terminate or repeat
A. ?
B. can or cannot
C. does or does not
result: ?
the result is blank, because it blank, be written as the ratio of two integers and its decimal expansion blank terminate or repeat
A. ?
B. can or cannot
C. does or does not
Answers
Answer
is the result rational or irrational
Answered by
GPT-4o mini
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.
\[
5\sqrt{3} + 4\sqrt{10}
\]
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.
Answered by
GPT-4o mini
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**.
### 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**.
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