Question

6.2 classwork/homework: simplifying nth roots
simplify:
\\(\sqrt{24}\\) \\(\sqrt3{-162}\\) \\(\sqrt3{729}\\)
\\(\sqrt{512}\\) \\(-6\sqrt4{80}\\) \\(-7\sqrt5{-96}\\)
\\(x^4\sqrt3{x^6}\\) \\(8x^2\sqrt3{54x^9}\\) \\(5x\sqrt4{48x^6}\\)
\\(3x^3\sqrt3{64x^5}\\) \\(3x^3\sqrt4{64x^5}\\) \\(3x^3\sqrt5{64x^5}\\)
\\(2x^2y^2\sqrt3{-108x^4y^6}\\) \\(4x^3y \sqrt{196x^4y^2}\\) \\(6x^3y^2z^3\sqrt4{80x^5y^4z^8}\\)

Explanation:

Step1: Factor into perfect powers

$\sqrt{24} = \sqrt{4 \times 6}$

Step2: Simplify the perfect square

$\sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

---

Step1: Factor into perfect cubes

$\sqrt[3]{-162} = \sqrt[3]{-27 \times 6}$

Step2: Simplify the perfect cube

$\sqrt[3]{-27} \times \sqrt[3]{6} = -3\sqrt[3]{6}$

---

Step1: Identify perfect cube

$\sqrt[3]{729} = \sqrt[3]{9^3}$

Step2: Simplify the cube root

$9$

---

Step1: Factor into perfect squares

$\sqrt{512} = \sqrt{256 \times 2}$

Step2: Simplify the perfect square

$\sqrt{256} \times \sqrt{2} = 16\sqrt{2}$

---

Step1: Factor into perfect 4th powers

$-6\sqrt[4]{80} = -6\sqrt[4]{16 \times 5}$

Step2: Simplify the perfect 4th root

$-6 \times 2\sqrt[4]{5} = -12\sqrt[4]{5}$

---

Step1: Factor into perfect 5th powers

$-7\sqrt[5]{-96} = -7\sqrt[5]{-32 \times 3}$

Step2: Simplify the perfect 5th root

$-7 \times (-2)\sqrt[5]{3} = 14\sqrt[5]{3}$

---

Step1: Combine exponents via rule $x^a \cdot x^b = x^{a+b}$

$x^4 \cdot x^{\frac{6}{3}} = x^4 \cdot x^2$

Step2: Add exponents

$x^{4+2} = x^6$

---

Step1: Simplify radical and variable

$8x^2\sqrt[3]{54x^9} = 8x^2 \cdot \sqrt[3]{27 \times 2} \cdot x^{\frac{9}{3}}$

Step2: Compute terms

$8x^2 \cdot 3\sqrt[3]{2} \cdot x^3 = 24x^{2+3}\sqrt[3]{2} = 24x^5\sqrt[3]{2}$

---

Step1: Factor into perfect 4th powers

$5x\sqrt[4]{48x^6} = 5x \cdot \sqrt[4]{16 \times 3} \cdot x^{\frac{6}{4}}$

Step2: Simplify terms

$5x \cdot 2\sqrt[4]{3} \cdot x^{\frac{3}{2}} = 10x^{1+\frac{3}{2}}\sqrt[4]{3} = 10x^{\frac{5}{2}}\sqrt[4]{3}$

---

Step1: Simplify radical and variable

$3x^3\sqrt[3]{64x^5} = 3x^3 \cdot 4 \cdot x^{\frac{5}{3}}$

Step2: Combine terms

$12x^{3+\frac{5}{3}} = 12x^{\frac{14}{3}}$

---

Step1: Factor into perfect 4th powers

$3x^3\sqrt[4]{64x^5} = 3x^3 \cdot \sqrt[4]{16 \times 4} \cdot x^{\frac{5}{4}}$

Step2: Simplify terms

$3x^3 \cdot 2\sqrt[4]{4} \cdot x^{\frac{5}{4}} = 6x^{3+\frac{5}{4}}\sqrt[4]{4} = 6x^{\frac{17}{4}}\sqrt[4]{4}$

---

Step1: Simplify radical and variable

$3x^3\sqrt[5]{64x^5} = 3x^3 \cdot \sqrt[5]{32 \times 2} \cdot x^{\frac{5}{5}}$

Step2: Combine terms

$3x^3 \cdot 2\sqrt[5]{2} \cdot x = 6x^{3+1}\sqrt[5]{2} = 6x^4\sqrt[5]{2}$

---

Step1: Factor into perfect cubes

$2x^2y^2\sqrt[3]{-108x^4y^6} = 2x^2y^2 \cdot \sqrt[3]{-27 \times 4} \cdot x^{\frac{4}{3}} \cdot y^{\frac{6}{3}}$

Step2: Simplify terms

$2x^2y^2 \cdot (-3)\sqrt[3]{4} \cdot x^{\frac{4}{3}} \cdot y^2 = -6x^{2+\frac{4}{3}}y^{2+2}\sqrt[3]{4} = -6x^{\frac{10}{3}}y^4\sqrt[3]{4}$

---

Step1: Simplify radical and variable

$4x^3y\sqrt{196x^4y^2} = 4x^3y \cdot 14 \cdot x^{\frac{4}{2}} \cdot y^{\frac{2}{2}}$

Step2: Combine terms

$56x^3y \cdot x^2 \cdot y = 56x^{3+2}y^{1+1} = 56x^5y^2$

---

Step1: Factor into perfect 4th powers

$6x^3y^2z^3\sqrt[4]{80x^5y^4z^8} = 6x^3y^2z^3 \cdot \sqrt[4]{16 \times 5} \cdot x^{\frac{5}{4}} \cdot y^{\frac{4}{4}} \cdot z^{\frac{8}{4}}$

Step2: Simplify terms

$6x^3y^2z^3 \cdot 2\sqrt[4]{5} \cdot x^{\frac{5}{4}} \cdot y \cdot z^2 = 12x^{3+\frac{5}{4}}y^{2+1}z^{3+2}\sqrt[4]{5} = 12x^{\frac{17}{4}}y^3z^5\sqrt[4]{5}$

Answer:

1. $2\sqrt{6}$
2. $-3\sqrt[3]{6}$
3. $9$
4. $16\sqrt{2}$
5. $-12\sqrt[4]{5}$
6. $14\sqrt[5]{3}$
7. $x^6$
8. $24x^5\sqrt[3]{2}$
9. $10x^{\frac{5}{2}}\sqrt[4]{3}$
10. $12x^{\frac{14}{3}}$
11. $6x^{\frac{17}{4}}\sqrt[4]{4}$
12. $6x^4\sqrt[5]{2}$
13. $-6x^{\frac{10}{3}}y^4\sqrt[3]{4}$
14. $56x^5y^2$
15. $12x^{\frac{17}{4}}y^3z^5\sqrt[4]{5}$