Question

6.3 classwork/homework: adding & subtracting nth root radicals

1. $4\sqrt{3} + \sqrt{27}$
2. $3\sqrt{50} - 2\sqrt{32}$
3. $- 3\sqrt{24} + 5\sqrt{54}$
4. $5\sqrt3{54} - 2\sqrt3{16}$
5. $7\sqrt3{162} - 5\sqrt3{32}$
6. $\sqrt3{96} + 2\sqrt3{3}$
7. $\sqrt3{108} + \sqrt3{192} + \sqrt3{500}$
8. $\sqrt4{1250} + \sqrt4{768} - \sqrt4{243}$
9. $\sqrt{5x^2} + x\sqrt{45}$
10. $\sqrt{8xy^2} + 2y\sqrt{18x}$
11. $2x\sqrt{32y} - \sqrt{18x^2y}$
12. $4\sqrt{32x^2} - 3x\sqrt{50}$

Explanation:

1. Problem 1: Simplify radicals

Rewrite $\sqrt{27}$ as $\sqrt{9 \times 3} = 3\sqrt{3}$
$4\sqrt{3} + 3\sqrt{3}$

1. Problem 1: Combine like radicals

$(4+3)\sqrt{3} = 7\sqrt{3}$

2. Problem 2: Simplify radicals

$3\sqrt{50}=3\sqrt{25 \times 2}=15\sqrt{2}$, $2\sqrt{32}=2\sqrt{16 \times 2}=8\sqrt{2}$
$15\sqrt{2} - 8\sqrt{2}$

2. Problem 2: Combine like radicals

$(15-8)\sqrt{2}=7\sqrt{2}$

3. Problem 3: Simplify radicals

$-3\sqrt{24}=-3\sqrt{4 \times 6}=-6\sqrt{6}$, $5\sqrt{54}=5\sqrt{9 \times 6}=15\sqrt{6}$
$-6\sqrt{6} + 15\sqrt{6}$

3. Problem 3: Combine like radicals

$(-6+15)\sqrt{6}=9\sqrt{6}$

4. Problem 4: Simplify radicals

$5\sqrt[3]{54}=5\sqrt[3]{27 \times 2}=15\sqrt[3]{2}$, $2\sqrt[3]{16}=2\sqrt[3]{8 \times 2}=4\sqrt[3]{2}$
$15\sqrt[3]{2} - 4\sqrt[3]{2}$

4. Problem 4: Combine like radicals

$(15-4)\sqrt[3]{2}=11\sqrt[3]{2}$

5. Problem 5: Simplify radicals

$7\sqrt[4]{162}=7\sqrt[4]{81 \times 2}=21\sqrt[4]{2}$, $5\sqrt[4]{32}=5\sqrt[4]{16 \times 2}=10\sqrt[4]{2}$
$21\sqrt[4]{2} - 10\sqrt[4]{2}$

5. Problem 5: Combine like radicals

$(21-10)\sqrt[4]{2}=11\sqrt[4]{2}$

6. Problem 6: Simplify radicals

$\sqrt[5]{96}=\sqrt[5]{32 \times 3}=2\sqrt[5]{3}$
$2\sqrt[5]{3} + 2\sqrt[5]{3}$

6. Problem 6: Combine like radicals

$(2+2)\sqrt[5]{3}=4\sqrt[5]{3}$

7. Problem 7: Simplify radicals

$\sqrt[3]{108}=\sqrt[3]{27 \times 4}=3\sqrt[3]{4}$, $\sqrt[3]{192}=\sqrt[3]{64 \times 3}=4\sqrt[3]{3}$, $\sqrt[3]{500}=\sqrt[3]{125 \times 4}=5\sqrt[3]{4}$
$3\sqrt[3]{4} + 4\sqrt[3]{3} + 5\sqrt[3]{4}$

7. Problem 7: Combine like radicals

$(3+5)\sqrt[3]{4} + 4\sqrt[3]{3}=8\sqrt[3]{4} + 4\sqrt[3]{3}$

8. Problem 8: Simplify radicals

$\sqrt[4]{1250}=\sqrt[4]{625 \times 2}=5\sqrt[4]{2}$, $\sqrt[4]{768}=\sqrt[4]{256 \times 3}=4\sqrt[4]{3}$, $\sqrt[4]{243}=\sqrt[4]{81 \times 3}=3\sqrt[4]{3}$
$5\sqrt[4]{2} + 4\sqrt[4]{3} - 3\sqrt[4]{3}$

8. Problem 8: Combine like radicals

$5\sqrt[4]{2} + (4-3)\sqrt[4]{3}=5\sqrt[4]{2} + \sqrt[4]{3}$

9. Problem 9: Simplify radicals

$\sqrt{5x^2}=x\sqrt{5}$, $x\sqrt{45}=x\sqrt{9 \times 5}=3x\sqrt{5}$
$x\sqrt{5} + 3x\sqrt{5}$

9. Problem 9: Combine like radicals

$(1+3)x\sqrt{5}=4x\sqrt{5}$

10. Problem 10: Simplify radicals

$\sqrt{8xy^2}=2y\sqrt{2x}$, $2y\sqrt{18x}=2y\sqrt{9 \times 2x}=6y\sqrt{2x}$
$2y\sqrt{2x} + 6y\sqrt{2x}$

10. Problem 10: Combine like radicals

$(2+6)y\sqrt{2x}=8y\sqrt{2x}$

11. Problem 11: Simplify radicals

$2x\sqrt{32y}=2x\sqrt{16 \times 2y}=8x\sqrt{2y}$, $\sqrt{18x^2y}=\sqrt{9x^2 \times 2y}=3x\sqrt{2y}$
$8x\sqrt{2y} - 3x\sqrt{2y}$

11. Problem 11: Combine like radicals

$(8-3)x\sqrt{2y}=5x\sqrt{2y}$

12. Problem 12: Simplify radicals

$4\sqrt{32x^2}=4\sqrt{16x^2 \times 2}=16x\sqrt{2}$, $3x\sqrt{50}=3x\sqrt{25 \times 2}=15x\sqrt{2}$
$16x\sqrt{2} - 15x\sqrt{2}$

12. Problem 12: Combine like radicals

$(16-15)x\sqrt{2}=x\sqrt{2}$

Answer:

1. $7\sqrt{3}$
2. $7\sqrt{2}$
3. $9\sqrt{6}$
4. $11\sqrt[3]{2}$
5. $11\sqrt[4]{2}$
6. $4\sqrt[5]{3}$
7. $8\sqrt[3]{4} + 4\sqrt[3]{3}$
8. $5\sqrt[4]{2} + \sqrt[4]{3}$
9. $4x\sqrt{5}$
10. $8y\sqrt{2x}$
11. $5x\sqrt{2y}$
12. $x\sqrt{2}$