Question
consider the expression 47/3 + 121 under a square root. will the sun of these numbers produce a rational or irrational number?
Answers
GPT-4o mini
To determine whether the expression \(\sqrt{\frac{47}{3} + 121}\) produces a rational or irrational number, we will first simplify the expression inside the square root.
1. First, convert \(121\) into a fraction with a denominator of \(3\):
\[
121 = \frac{121 \times 3}{3} = \frac{363}{3}
\]
2. Next, add \(\frac{47}{3}\) and \(\frac{363}{3}\):
\[
\frac{47}{3} + \frac{363}{3} = \frac{47 + 363}{3} = \frac{410}{3}
\]
3. Now, we can write the expression under the square root:
\[
\sqrt{\frac{410}{3}}
\]
4. We can further simplify this:
\[
\sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}}
\]
To determine if \(\sqrt{\frac{410}{3}}\) is rational or irrational, we need to investigate \(\sqrt{410}\) and \(\sqrt{3}\):
- \(\sqrt{3}\) is known to be an irrational number.
- For \(\sqrt{410}\), we need to check if \(410\) is a perfect square. The prime factorization of \(410\) is:
\[
410 = 2 \times 5 \times 41
\]
Since \(410\) does not have any factor raised to an even power, \(\sqrt{410}\) is also an irrational number.
Since the product of two irrational numbers (as long as they are not precisely structured to yield a rational product) will typically result in an irrational number, and given that both \(\sqrt{410}\) and \(\sqrt{3}\) are irrational, we conclude:
\[
\sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}} \text{ is also irrational.}
\]
Thus, the expression \(\sqrt{\frac{47}{3} + 121}\) produces an irrational number.
1. First, convert \(121\) into a fraction with a denominator of \(3\):
\[
121 = \frac{121 \times 3}{3} = \frac{363}{3}
\]
2. Next, add \(\frac{47}{3}\) and \(\frac{363}{3}\):
\[
\frac{47}{3} + \frac{363}{3} = \frac{47 + 363}{3} = \frac{410}{3}
\]
3. Now, we can write the expression under the square root:
\[
\sqrt{\frac{410}{3}}
\]
4. We can further simplify this:
\[
\sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}}
\]
To determine if \(\sqrt{\frac{410}{3}}\) is rational or irrational, we need to investigate \(\sqrt{410}\) and \(\sqrt{3}\):
- \(\sqrt{3}\) is known to be an irrational number.
- For \(\sqrt{410}\), we need to check if \(410\) is a perfect square. The prime factorization of \(410\) is:
\[
410 = 2 \times 5 \times 41
\]
Since \(410\) does not have any factor raised to an even power, \(\sqrt{410}\) is also an irrational number.
Since the product of two irrational numbers (as long as they are not precisely structured to yield a rational product) will typically result in an irrational number, and given that both \(\sqrt{410}\) and \(\sqrt{3}\) are irrational, we conclude:
\[
\sqrt{\frac{410}{3}} = \frac{\sqrt{410}}{\sqrt{3}} \text{ is also irrational.}
\]
Thus, the expression \(\sqrt{\frac{47}{3} + 121}\) produces an irrational number.