Question

multiply. write your answer in simplest form.\\(sqrt{245} cdot sqrt{7}\\)

Explanation:

Step1: Use property of square roots

Recall that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ for $a\geq0, b\geq0$. So, $\sqrt{245} \cdot \sqrt{7} = \sqrt{245 \times 7}$.

Step2: Calculate the product inside the square root

Calculate $245\times7 = 1715$? Wait, no, wait, 245 is $49\times5$, so 245×7 = 49×5×7 = 49×35. Wait, no, wait, 245×7: 245×7 = (200 + 45)×7 = 1400 + 315 = 1715? Wait, no, that's wrong. Wait, 245 is 49×5, so 245×7 = 49×5×7 = 49×35. But also, 49 is 7². Wait, maybe factor 245 first. 245 = 49×5 = 7²×5. So $\sqrt{245} = \sqrt{7^2 \times 5} = 7\sqrt{5}$. Then $\sqrt{245} \cdot \sqrt{7} = 7\sqrt{5} \cdot \sqrt{7} = 7\sqrt{35}$. Wait, but using the first method: $\sqrt{245 \times 7} = \sqrt{1715}$? Wait, no, 245×7: 245×7 = 1715? Wait, 250×7 = 1750, so 245×7 = 1750 - 5×7 = 1750 - 35 = 1715. But 1715 can be factored: 1715 ÷ 5 = 343, and 343 is 7³. So 1715 = 5×7³ = 5×7²×7. So $\sqrt{1715} = \sqrt{7^2 \times 35} = 7\sqrt{35}$. Ah, there we go. So either way, we get $7\sqrt{35}$? Wait, no, wait, let's do it again. Wait, $\sqrt{245} \cdot \sqrt{7} = \sqrt{245 \times 7} = \sqrt{1715}$. Now, factor 1715: 1715 ÷ 5 = 343, 343 ÷ 7 = 49, 49 ÷ 7 = 7. So 1715 = 5×7×49 = 5×7×7² = 7²×35. So $\sqrt{1715} = \sqrt{7^2 \times 35} = 7\sqrt{35}$. Alternatively, factor 245 as 49×5, so $\sqrt{245} = 7\sqrt{5}$, then multiply by $\sqrt{7}$: $7\sqrt{5} \cdot \sqrt{7} = 7\sqrt{35}$. Yep, that's correct.

Wait, but maybe I made a mistake in the first multiplication. Wait, 245×7: 245×7. Let's calculate 200×7=1400, 40×7=280, 5×7=35. So 1400+280=1680, 1680+35=1715. Correct. Then 1715=5×343=5×7³=5×7²×7. So $\sqrt{1715}=\sqrt{7²×5×7}=\sqrt{7²×35}=7\sqrt{35}$. So that's the simplest form.

Wait, but let's check with the property again. $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$. So $\sqrt{245}\cdot\sqrt{7}=\sqrt{245\times7}=\sqrt{1715}$. Then factor 1715: 1715 ÷ 5 = 343, 343 is 7³, so 1715 = 5×7³ = 5×7²×7. So $\sqrt{5×7²×7}=\sqrt{7²×35}=7\sqrt{35}$. Yes, that's correct.

Answer:

$7\sqrt{35}$