Question

1. $8sqrt{12y} - 4sqrt{27y}$ 8. $3sqrt3{5} + 6sqrt3{5}$ 11. $4sqrt3{32x^3} - sqrt3{108x^3}$ 12. $sqrt3{54x^5} + 2sqrt3{8x^6}$

Explanation:

Step1: Simplify radicals (Problem7)

$8\sqrt{12y}=8\sqrt{4\cdot3y}=8\cdot2\sqrt{3y}=16\sqrt{3y}$
$4\sqrt{27y}=4\sqrt{9\cdot3y}=4\cdot3\sqrt{3y}=12\sqrt{3y}$

Step2: Subtract like radicals (Problem7)

$16\sqrt{3y}-12\sqrt{3y}=(16-12)\sqrt{3y}$

Step3: Combine like radicals (Problem8)

$3\sqrt[3]{5}+6\sqrt[3]{5}=(3+6)\sqrt[3]{5}$

Step4: Simplify radicals (Problem11)

$4\sqrt[3]{32x^3}=4\sqrt[3]{8x^3\cdot4}=4\cdot2x\sqrt[3]{4}=8x\sqrt[3]{4}$
$\sqrt[3]{108x^3}=\sqrt[3]{27x^3\cdot4}=3x\sqrt[3]{4}$

Step5: Subtract like radicals (Problem11)

$8x\sqrt[3]{4}-3x\sqrt[3]{4}=(8x-3x)\sqrt[3]{4}$

Step6: Simplify radicals (Problem12)

$\sqrt[3]{54x^5}=\sqrt[3]{27x^3\cdot2x^2}=3x\sqrt[3]{2x^2}$
$2\sqrt[3]{8x^6}=2\cdot2x^2=4x^2$

Step7: Combine terms (Problem12)

$3x\sqrt[3]{2x^2}+4x^2$

Answer:

1. $4\sqrt{3y}$
2. $9\sqrt[3]{5}$
3. $5x\sqrt[3]{4}$
4. $3x\sqrt[3]{2x^2}+4x^2$