Question

subtracting radicals

1. $-sqrt{5} - sqrt{5}$
2. $sqrt{48} - sqrt{27}$
3. $sqrt{5} - sqrt{80}$
4. $-3sqrt{10} - 4sqrt{40}$

answers
$-11sqrt{10}$
$-2sqrt{5}$
$-48sqrt{2}$
$sqrt{3}$
$-3sqrt{5}$

Explanation:

Step1: Combine like radicals

$-\sqrt{5} - \sqrt{5} = (-1-1)\sqrt{5} = -2\sqrt{5}$

Step2: Simplify each radical first

$\sqrt{48} = \sqrt{16\times3} = 4\sqrt{3}$, $\sqrt{27} = \sqrt{9\times3} = 3\sqrt{3}$
Subtract: $4\sqrt{3} - 3\sqrt{3} = \sqrt{3}$

Step3: Simplify the second radical

$\sqrt{80} = \sqrt{16\times5} = 4\sqrt{5}$
Subtract: $\sqrt{5} - 4\sqrt{5} = (1-4)\sqrt{5} = -3\sqrt{5}$

Step4: Simplify the second radical

$\sqrt{40} = \sqrt{4\times10} = 2\sqrt{10}$
Rewrite and combine: $-3\sqrt{10} - 4\times2\sqrt{10} = -3\sqrt{10} - 8\sqrt{10} = (-3-8)\sqrt{10} = -11\sqrt{10}$

Answer:

1. $-2\sqrt{5}$
2. $\sqrt{3}$
3. $-3\sqrt{5}$
4. $-11\sqrt{10}$