Mixed Exercises Simplify. \( \begin{array}{ll}\text { 39. } \sqrt[n]{16 y^{4} z^{12}} & \text { 40. } \sqrt[3]{-54 x^{6} y^{11}} \\ \text { 41. } \frac{x+1}{\sqrt{x-1}} & \text { 42. } \frac{x-2}{\sqrt{x^{2}-4}} \\ \text { 43. } 3 \sqrt{24 x}-2 \sqrt{54 x}+\sqrt{48} & \text { 44. } 5 \sqrt{18 c}+3 \sqrt{72 c}+6 \sqrt{76} \\ \text { 45. } 10 \sqrt{175 a}-4 \sqrt{112 a}-2 \sqrt{63 a} & \text { 46. } 7 \sqrt{204 y}+4 \sqrt{459 y}-8 \sqrt{140 y} \\ \text { 47. VOLUME McKenzie has a rectangular orism with dimensions } 20 \text { inches by } 35 \text { inches }\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-2\right)}{\left(x^{2}-4\right)^{\frac{1}{2}}}\) - step1: Remove the parentheses: \(\frac{x-2}{\left(x^{2}-4\right)^{\frac{1}{2}}}\) - step2: Simplify: \(\frac{x-2}{\sqrt{x^{2}-4}}\) - step3: Calculate: \(\frac{\sqrt{x^{2}-4}}{x+2}\) Calculate or simplify the expression \( 5 * (18c)^(1/2) + 3 * (72c)^(1/2) + 6 * (76)^(1/2) \). Simplify the expression by following steps: - step0: Solution: \(5\left(18c\right)^{\frac{1}{2}}+3\left(72c\right)^{\frac{1}{2}}+6\times 76^{\frac{1}{2}}\) - step1: Multiply the terms: \(5\times 18^{\frac{1}{2}}c^{\frac{1}{2}}+3\left(72c\right)^{\frac{1}{2}}+6\times 76^{\frac{1}{2}}\) - step2: Multiply the terms: \(5\times 18^{\frac{1}{2}}c^{\frac{1}{2}}+3\times 72^{\frac{1}{2}}c^{\frac{1}{2}}+6\times 76^{\frac{1}{2}}\) - step3: Add the terms: \(33\times 2^{\frac{1}{2}}c^{\frac{1}{2}}+6\times 76^{\frac{1}{2}}\) - step4: Simplify: \(33\sqrt{2c}+6\times 76^{\frac{1}{2}}\) - step5: Simplify: \(33\sqrt{2c}+6\times 2\sqrt{19}\) - step6: Expand the expression: \(33\sqrt{2c}+12\sqrt{19}\) Calculate or simplify the expression \( (-54)^(1/3) * (x^(6/3)) * (y^(11/3)) \). Simplify the expression by following steps: - step0: Solution: \(\left(-54\right)^{\frac{1}{3}}x^{\frac{6}{3}}\left(y^{\frac{11}{3}}\right)\) - step1: Rewrite the expression: \(\left(-54\right)^{\frac{1}{3}}x^{\frac{6}{3}}y^{\frac{11}{3}}\) - step2: Divide the terms: \(\left(-54\right)^{\frac{1}{3}}x^{2}y^{\frac{11}{3}}\) - step3: Simplify: \(-54^{\frac{1}{3}}x^{2}y^{\frac{11}{3}}\) - step4: Simplify: \(-3\sqrt[3]{2}\times x^{2}y^{\frac{11}{3}}\) - step5: Calculate: \(-3x^{2}y^{3}\sqrt[3]{2y^{2}}\) Calculate or simplify the expression \( 7 * (204y)^(1/2) + 4 * (459y)^(1/2) - 8 * (140y)^(1/2) \). Simplify the expression by following steps: - step0: Solution: \(7\left(204y\right)^{\frac{1}{2}}+4\left(459y\right)^{\frac{1}{2}}-8\left(140y\right)^{\frac{1}{2}}\) - step1: Multiply the terms: \(7\times 204^{\frac{1}{2}}y^{\frac{1}{2}}+4\left(459y\right)^{\frac{1}{2}}-8\left(140y\right)^{\frac{1}{2}}\) - step2: Multiply the terms: \(7\times 204^{\frac{1}{2}}y^{\frac{1}{2}}+4\times 459^{\frac{1}{2}}y^{\frac{1}{2}}-8\left(140y\right)^{\frac{1}{2}}\) - step3: Multiply the terms: \(7\times 204^{\frac{1}{2}}y^{\frac{1}{2}}+4\times 459^{\frac{1}{2}}y^{\frac{1}{2}}-8\times 140^{\frac{1}{2}}y^{\frac{1}{2}}\) - step4: Simplify: \(14\sqrt{51y}+4\times 459^{\frac{1}{2}}y^{\frac{1}{2}}-8\times 140^{\frac{1}{2}}y^{\frac{1}{2}}\) - step5: Simplify: \(14\sqrt{51y}+12\sqrt{51y}-8\times 140^{\frac{1}{2}}y^{\frac{1}{2}}\) - step6: Simplify: \(14\sqrt{51y}+12\sqrt{51y}-16\sqrt{35y}\) - step7: Add the terms: \(26\sqrt{51y}-16\sqrt{35y}\) Calculate or simplify the expression \( (16)^(1/n) * (y^(4/n)) * (z^(12/n)) \). Simplify the expression by following steps: - step0: Solution: \(16^{\frac{1}{n}}y^{\frac{4}{n}}z^{\frac{12}{n}}\) - step1: Multiply the terms: \(\left(2y\right)^{\frac{4}{n}}z^{\frac{12}{n}}\) - step2: Multiply the exponents: \(\left(16y^{4}\right)^{\frac{1}{n}}\left(z^{12}\right)^{\frac{1}{n}}\) - step3: Calculate: \(\left(16y^{4}z^{12}\right)^{\frac{1}{n}}\) - step4: Calculate: \(\left(2yz^{3}\right)^{\frac{4}{n}}\) Calculate or simplify the expression \( 10 * (175a)^(1/2) - 4 * (112a)^(1/2) - 2 * (63a)^(1/2) \). Simplify the expression by following steps: - step0: Solution: \(10\left(175a\right)^{\frac{1}{2}}-4\left(112a\right)^{\frac{1}{2}}-2\left(63a\right)^{\frac{1}{2}}\) - step1: Multiply the terms: \(10\times 175^{\frac{1}{2}}a^{\frac{1}{2}}-4\left(112a\right)^{\frac{1}{2}}-2\left(63a\right)^{\frac{1}{2}}\) - step2: Multiply the terms: \(10\times 175^{\frac{1}{2}}a^{\frac{1}{2}}-4\times 112^{\frac{1}{2}}a^{\frac{1}{2}}-2\left(63a\right)^{\frac{1}{2}}\) - step3: Multiply the terms: \(10\times 175^{\frac{1}{2}}a^{\frac{1}{2}}-4\times 112^{\frac{1}{2}}a^{\frac{1}{2}}-2\times 63^{\frac{1}{2}}a^{\frac{1}{2}}\) - step4: Collect like terms: \(\left(10\times 175^{\frac{1}{2}}-4\times 112^{\frac{1}{2}}-2\times 63^{\frac{1}{2}}\right)a^{\frac{1}{2}}\) - step5: Subtract the numbers: \(28\times 7^{\frac{1}{2}}a^{\frac{1}{2}}\) - step6: Simplify: \(28\sqrt{7}\times a^{\frac{1}{2}}\) - step7: Calculate: \(28\sqrt{7a}\) Calculate or simplify the expression \( 3 * (24x)^(1/2) - 2 * (54x)^(1/2) + (48)^(1/2) \). Simplify the expression by following steps: - step0: Solution: \(3\left(24x\right)^{\frac{1}{2}}-2\left(54x\right)^{\frac{1}{2}}+48^{\frac{1}{2}}\) - step1: Multiply the terms: \(3\times 24^{\frac{1}{2}}x^{\frac{1}{2}}-2\left(54x\right)^{\frac{1}{2}}+48^{\frac{1}{2}}\) - step2: Multiply the terms: \(3\times 24^{\frac{1}{2}}x^{\frac{1}{2}}-2\times 54^{\frac{1}{2}}x^{\frac{1}{2}}+48^{\frac{1}{2}}\) - step3: Simplify: \(6\sqrt{6x}-2\times 54^{\frac{1}{2}}x^{\frac{1}{2}}+48^{\frac{1}{2}}\) - step4: Simplify: \(6\sqrt{6x}-6\sqrt{6x}+48^{\frac{1}{2}}\) - step5: Simplify: \(6\sqrt{6x}-6\sqrt{6x}+4\sqrt{3}\) - step6: Collect like terms: \(4\sqrt{3}\) Calculate or simplify the expression \( (x+1)/(x-1)^(1/2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+1\right)}{\left(x-1\right)^{\frac{1}{2}}}\) - step1: Remove the parentheses: \(\frac{x+1}{\left(x-1\right)^{\frac{1}{2}}}\) - step2: Simplify: \(\frac{x+1}{\sqrt{x-1}}\) - step3: Calculate: \(\frac{x\sqrt{x-1}+\sqrt{x-1}}{x-1}\) Here are the simplified results for each of the exercises: **39.** Simplifying \( \sqrt[n]{16 y^{4} z^{12}} \): \[ \sqrt[n]{16 y^{4} z^{12}} = (2yz^{3})^{\frac{4}{n}} \] **40.** Simplifying \( \sqrt[3]{-54 x^{6} y^{11}} \): \[ \sqrt[3]{-54 x^{6} y^{11}} = -3x^{2}y^{3}\sqrt[3]{2y^{2}} \] **41.** Simplifying \( \frac{x+1}{\sqrt{x-1}} \): \[ \frac{x+1}{\sqrt{x-1}} = \frac{x\sqrt{x-1} + \sqrt{x-1}}{x-1} \] **42.** Simplifying \( \frac{x-2}{\sqrt{x^{2}-4}} \): \[ \frac{x-2}{\sqrt{x^{2}-4}} = \frac{\sqrt{x^{2}-4}}{x+2} \] **43.** Simplifying \( 3 \sqrt{24 x}-2 \sqrt{54 x}+\sqrt{48} \): \[ 3 \sqrt{24 x}-2 \sqrt{54 x}+\sqrt{48} = 4\sqrt{3} \] **44.** Simplifying \( 5 \sqrt{18 c}+3 \sqrt{72 c}+6 \sqrt{76} \): \[ 5 \sqrt{18 c}+3 \sqrt{72 c}+6 \sqrt{76} = 33\sqrt{2c}+12\sqrt{19} \] **45.** Simplifying \( 10 \sqrt{175 a}-4 \sqrt{112 a}-2 \sqrt{63 a} \): \[ 10 \sqrt{175 a}-4 \sqrt{112 a}-2 \sqrt{63 a} = 28\sqrt{7a} \] **46.** Simplifying \( 7 \sqrt{204 y}+4 \sqrt{459 y}-8 \sqrt{140 y} \): \[ 7 \sqrt{204 y}+4 \sqrt{459 y}-8 \sqrt{140 y} = 26\sqrt{51y}-16\sqrt{35y} \] If you have any further questions or need additional assistance, feel free to ask!