Question
Prove that the product of two rational numbers radical 36 and 5/3 is a rational number find the product and explain how you know it is a rational number
Answers
GPT-4o mini
To prove that the product of the two rational numbers \(\sqrt{36}\) and \(\frac{5}{3}\) is a rational number, we will first calculate the product and then explain why it is rational.
1. **Calculate \(\sqrt{36}\)**:
\[
\sqrt{36} = 6
\]
This is because \(6 \times 6 = 36\).
2. **Calculate the product**:
Now we need to multiply \(6\) by \(\frac{5}{3}\):
\[
6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10
\]
3. **Check if the result is a rational number**:
A rational number is defined as a number that can be expressed as the quotient of two integers (i.e., in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\)). The result we obtained is \(10\).
We can express \(10\) as:
\[
10 = \frac{10}{1}
\]
Here, \(10\) is an integer (which can be considered as \(\frac{10}{1}\)), and \(1\) is also an integer, with \(1 \neq 0\).
Hence, since \(10\) can be expressed as the quotient of two integers, we conclude that \(10\) is indeed a rational number.
**Conclusion**: The product of \(\sqrt{36}\) and \(\frac{5}{3}\) is \(10\), which is a rational number.
1. **Calculate \(\sqrt{36}\)**:
\[
\sqrt{36} = 6
\]
This is because \(6 \times 6 = 36\).
2. **Calculate the product**:
Now we need to multiply \(6\) by \(\frac{5}{3}\):
\[
6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10
\]
3. **Check if the result is a rational number**:
A rational number is defined as a number that can be expressed as the quotient of two integers (i.e., in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\)). The result we obtained is \(10\).
We can express \(10\) as:
\[
10 = \frac{10}{1}
\]
Here, \(10\) is an integer (which can be considered as \(\frac{10}{1}\)), and \(1\) is also an integer, with \(1 \neq 0\).
Hence, since \(10\) can be expressed as the quotient of two integers, we conclude that \(10\) is indeed a rational number.
**Conclusion**: The product of \(\sqrt{36}\) and \(\frac{5}{3}\) is \(10\), which is a rational number.