To find the difference \( \sqrt{8125} - \sqrt{780} \), we first need to calculate the square roots of each number.
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Calculate \( \sqrt{8125} \): \[ \sqrt{8125} = \sqrt{25 \times 325} = \sqrt{25} \times \sqrt{325} = 5\sqrt{325} \] Next, simplify \( \sqrt{325} \): \[ \sqrt{325} = \sqrt{25 \times 13} = \sqrt{25} \times \sqrt{13} = 5\sqrt{13} \] Therefore: \[ \sqrt{8125} = 5 \times 5\sqrt{13} = 25\sqrt{13} \]
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Calculate \( \sqrt{780} \): \[ \sqrt{780} = \sqrt{4 \times 195} = \sqrt{4} \times \sqrt{195} = 2\sqrt{195} \] Next, simplify \( \sqrt{195} \): \[ \sqrt{195} = \sqrt{39 \times 5} \] Since 39 does not have a perfect square factor, we keep it as is: \[ \sqrt{780} = 2\sqrt{195} \]
Now, we have: \[ \sqrt{8125} - \sqrt{780} = 25\sqrt{13} - 2\sqrt{195} \]
This is the simplified form, as \( 25\sqrt{13} \) and \( 2\sqrt{195} \) cannot be combined further since they are not like terms. If you want a numerical approximation:
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Calculate \( \sqrt{13} \approx 3.60555 \) \[ 25\sqrt{13} \approx 25 \times 3.60555 \approx 90.13875 \]
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Calculate \( \sqrt{195} \approx 13.96424 \) \[ 2\sqrt{195} \approx 2 \times 13.96424 \approx 27.92848 \]
Now, calculating the difference: \[ \sqrt{8125} - \sqrt{780} \approx 90.13875 - 27.92848 \approx 62.21027 \]
Thus, the difference \( \sqrt{8125} - \sqrt{780} \) is approximately 62.21.
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