Question

homework
7.01: swbat perform operations with radicals and rationalize the denominator
show all work and annotations for full credit.
**#1.) simplify the following expression:
$7x + 3y + 3(2x - 4y) - 11$
$7x + 3y + 6x - 12y - 11$
$13x - 9y - 11$
**#2.) express in simplest radical form:
$\frac{sqrt{8}}{sqrt{3}}$
$\frac{sqrt{8}}{sqrt{3}} \times \frac{sqrt{3}}{sqrt{3}} = \frac{sqrt{24}}{3}$
$sqrt{24}$
**#3.) express in simplest radical form
$sqrt{68} + sqrt{75} - sqrt{153} + sqrt{27}$
**#4.) the length of one side of the square below is $sqrt{20}$. determine the perimeter of the square in simplest radical form.

Explanation:

#1 Step1: Distribute the 3

$7x + 3y + 3(2x - 4y) - 11 = 7x + 3y + 6x - 12y - 11$

#1 Step2: Combine like x terms

$7x + 6x + 3y - 12y - 11 = 13x + 3y - 12y - 11$

#1 Step3: Combine like y terms

$13x + (3y - 12y) - 11 = 13x - 9y - 11$

#2 Step1: Rationalize the denominator

$\frac{\sqrt{8}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{8 \times 3}}{3} = \frac{\sqrt{24}}{3}$

#2 Step2: Simplify $\sqrt{24}$

$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$, so $\frac{2\sqrt{6}}{3}$

#3 Step1: Simplify each radical

$\sqrt{68} = \sqrt{4 \times 17} = 2\sqrt{17}$
$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$
$\sqrt{153} = \sqrt{9 \times 17} = 3\sqrt{17}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

#3 Step2: Substitute back and combine like radicals

$2\sqrt{17} + 5\sqrt{3} - 3\sqrt{17} + 3\sqrt{3} = (2\sqrt{17} - 3\sqrt{17}) + (5\sqrt{3} + 3\sqrt{3})$

#3 Step3: Calculate combined terms

$-\sqrt{17} + 8\sqrt{3}$

#4 Step1: Recall square perimeter formula

Perimeter $P = 4 \times \text{side length}$, so $P = 4\sqrt{20}$

#4 Step2: Simplify $\sqrt{20}$

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$, so $4 \times 2\sqrt{5} = 8\sqrt{5}$

Answer:

1. $13x - 9y - 11$
2. $\frac{2\sqrt{6}}{3}$
3. $8\sqrt{3} - \sqrt{17}$
4. $8\sqrt{5}$