Question
Multiply and Divide Radicals Quick Check
3 of 53 of 5 Items
Question
Divide 410√2√ . Provide an exact answer. (1 point)
Responses
45–√
4 start root 5 end root
25–√
2 start root 5 end root
210−−√
2 start root 10 end root
This expression is already simplified and cannot be divided further.
Answers
GPT-4o mini
To simplify \( \frac{410\sqrt{2}}{\sqrt{5}} \), we first rewrite the expression as follows:
\[
\frac{410\sqrt{2}}{\sqrt{5}} = 410\cdot \frac{\sqrt{2}}{\sqrt{5}} = 410\cdot \sqrt{\frac{2}{5}}
\]
Next, we can simplify \( \sqrt{\frac{2}{5}} \):
\[
\sqrt{\frac{2}{5}} = \frac{\sqrt{2}}{\sqrt{5}}
\]
Multiplying the numerator and denominator by \( \sqrt{5} \) to rationalize the denominator:
\[
\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{10}}{5}
\]
Now we substitute this back into our expression:
\[
410 \cdot \sqrt{\frac{2}{5}} = 410 \cdot \frac{\sqrt{10}}{5} = \frac{410\sqrt{10}}{5} = 82\sqrt{10}
\]
Thus, the final simplified answer is:
\[
82\sqrt{10}
\]
Since none of the provided answers directly match \( 82\sqrt{10} \), please double-check the expression that needs to be simplified. Based on the operations shown above, \( 82\sqrt{10} \) is indeed the correct simplification. If "210−−√" refers to a potential answer, it might have been intended to show \( 210\sqrt{10} \) or something similar, which also would not match.
If the original problem points to a specific format, please let me know, and we can explore alternatives.
\[
\frac{410\sqrt{2}}{\sqrt{5}} = 410\cdot \frac{\sqrt{2}}{\sqrt{5}} = 410\cdot \sqrt{\frac{2}{5}}
\]
Next, we can simplify \( \sqrt{\frac{2}{5}} \):
\[
\sqrt{\frac{2}{5}} = \frac{\sqrt{2}}{\sqrt{5}}
\]
Multiplying the numerator and denominator by \( \sqrt{5} \) to rationalize the denominator:
\[
\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{10}}{5}
\]
Now we substitute this back into our expression:
\[
410 \cdot \sqrt{\frac{2}{5}} = 410 \cdot \frac{\sqrt{10}}{5} = \frac{410\sqrt{10}}{5} = 82\sqrt{10}
\]
Thus, the final simplified answer is:
\[
82\sqrt{10}
\]
Since none of the provided answers directly match \( 82\sqrt{10} \), please double-check the expression that needs to be simplified. Based on the operations shown above, \( 82\sqrt{10} \) is indeed the correct simplification. If "210−−√" refers to a potential answer, it might have been intended to show \( 210\sqrt{10} \) or something similar, which also would not match.
If the original problem points to a specific format, please let me know, and we can explore alternatives.