Question

lesson 4: operations with radicals - (addition & subtraction)
do now: write each answer in simplest radical form;

1. $-sqrt{2}(3sqrt{2})$
2. $\frac{8sqrt{12}}{2sqrt{3}}$

steps for addition & subtraction:

1. simplify all radicals
2. combine the coefficients with identical numbers under the square root

example 1: $2sqrt{7} + 5sqrt{3} - 4sqrt{3}$
example 2: $6sqrt{50} - 2sqrt{8}$
practice:

1. $7sqrt{2} + 6sqrt{2}$
2. $12sqrt{5} - 8sqrt{5}$
3. $3sqrt{3} + sqrt{12}$
4. $sqrt{18} - sqrt{2}$
5. $9sqrt{8} - sqrt{50}$
6. $-sqrt{18} - sqrt{98}$

Explanation:

Step1: Simplify $\boldsymbol{-\sqrt{2}(3\sqrt{2})}$

Multiply coefficients and radicals:
$-\sqrt{2}(3\sqrt{2}) = -3 \times \sqrt{2 \times 2} = -3 \times 2 = -6$

Step2: Simplify $\boldsymbol{\frac{8\sqrt{12}}{2\sqrt{3}}}$

Simplify coefficients and radicals:
$\frac{8}{2} \times \frac{\sqrt{12}}{\sqrt{3}} = 4 \times \sqrt{\frac{12}{3}} = 4 \times \sqrt{4} = 4 \times 2 = 8$

Step3: Combine $\boldsymbol{7\sqrt{2} + 6\sqrt{2}}$

Add like radical coefficients:
$(7+6)\sqrt{2} = 13\sqrt{2}$

Step4: Combine $\boldsymbol{12\sqrt{5} - 8\sqrt{5}}$

Subtract like radical coefficients:
$(12-8)\sqrt{5} = 4\sqrt{5}$

Step5: Simplify $\boldsymbol{3\sqrt{3} + \sqrt{12}}$

Rewrite $\sqrt{12}$, then combine:
$3\sqrt{3} + \sqrt{4 \times 3} = 3\sqrt{3} + 2\sqrt{3} = (3+2)\sqrt{3} = 5\sqrt{3}$

Step6: Simplify $\boldsymbol{\sqrt{18} - \sqrt{2}}$

Rewrite $\sqrt{18}$, then combine:
$\sqrt{9 \times 2} - \sqrt{2} = 3\sqrt{2} - \sqrt{2} = (3-1)\sqrt{2} = 2\sqrt{2}$

Step7: Simplify $\boldsymbol{9\sqrt{8} - \sqrt{50}}$

Rewrite radicals, then combine:
$9\sqrt{4 \times 2} - \sqrt{25 \times 2} = 9 \times 2\sqrt{2} - 5\sqrt{2} = 18\sqrt{2} - 5\sqrt{2} = (18-5)\sqrt{2} = 13\sqrt{2}$

Step8: Simplify $\boldsymbol{-\sqrt{18} - \sqrt{98}}$

Rewrite radicals, then combine:
$-\sqrt{9 \times 2} - \sqrt{49 \times 2} = -3\sqrt{2} - 7\sqrt{2} = (-3-7)\sqrt{2} = -10\sqrt{2}$

Answer:

1. $-6$
2. $8$
3. $13\sqrt{2}$
4. $4\sqrt{5}$
5. $5\sqrt{3}$
6. $2\sqrt{2}$
7. $13\sqrt{2}$
8. $-10\sqrt{2}$