Question
is √350 irrational
Answers
GPT-4o mini
To determine if \(\sqrt{350}\) is rational or irrational, we can start by simplifying the square root.
First, we can factor 350:
\[
350 = 2 \times 5^2 \times 7
\]
Now, we can rewrite the square root:
\[
\sqrt{350} = \sqrt{2 \times 5^2 \times 7} = \sqrt{5^2} \times \sqrt{2} \times \sqrt{7} = 5 \times \sqrt{14}
\]
This shows that \(\sqrt{350} = 5\sqrt{14}\).
Since \(\sqrt{14}\) is not a perfect square (14 is not a perfect square), \(\sqrt{14}\) is irrational. Therefore, \(5\sqrt{14}\) is also irrational because the product of a rational number (5) and an irrational number (\(\sqrt{14}\)) is irrational.
Thus, \(\sqrt{350}\) is irrational.
First, we can factor 350:
\[
350 = 2 \times 5^2 \times 7
\]
Now, we can rewrite the square root:
\[
\sqrt{350} = \sqrt{2 \times 5^2 \times 7} = \sqrt{5^2} \times \sqrt{2} \times \sqrt{7} = 5 \times \sqrt{14}
\]
This shows that \(\sqrt{350} = 5\sqrt{14}\).
Since \(\sqrt{14}\) is not a perfect square (14 is not a perfect square), \(\sqrt{14}\) is irrational. Therefore, \(5\sqrt{14}\) is also irrational because the product of a rational number (5) and an irrational number (\(\sqrt{14}\)) is irrational.
Thus, \(\sqrt{350}\) is irrational.