simplify each expression.
59. $sqrt{245m^{9}n^{5}}$
60. $2sqrt{5} cdot 7sqrt{10}$
61. $sqrt{\frac{96d^{4}e^{2}f^{8}}{75de^{6}}}$
62. $-11sqrt3{250}$
63. $sqrt3{320x^{14}y^{17}z^{20}}$
64. $sqrt3{45k^{4}m^{10}} cdot sqrt3{32k^{7}m^{3}}$
65. $sqrt3{\frac{264w^{6}xy^{7}}{3w^{6}x^{4}y^{3}}}$
66. $sqrt3{\frac{48a^{7}}{125b^{9}}}$
67. area the base of a triangle measures $6sqrt{2}$ meters and the height measures $3sqrt{6}$ meters. what is the area?
68. area the length of a rectangle measures $8sqrt{12}$ centimeters and the width measures $4sqrt{8}$ centimeters. what is the area of the rectangle?
69. mean the geometric mean of two numbers $h$ and $k$ can be found by evaluating $sqrt{h cdot k}$. find the geometric mean of 32 and 14 in simplified radical form.
70. theatre crew fernanda oversees painting props for a theatre production. she needs to create a cube that will be used as a prop by the actors. she has enough paint to cover 5 square yards. the formula $s = sqrt{\frac{a}{6}}$ gives the longest side length $s$ in yards of a cubic prop fernanda can make, where $a$ is the surface area to be covered with paint. what is the longest side length of a cube fernanda could make and not have to purchase any more paint? round to the nearest tenth of a yard.
71. scale model while on vacation, deon decided to purchase a scale model of the empire state building. before making the purchase, deon wants to determine the maximum height of a model that will fit inside his suitcase. deons suitcase measures 22 inches long by 14 inches wide by 9 inches tall. the diagonal length $d$ is given by $d = sqrt{l^{2} + w^{2} + h^{2}}$, where $l$ is the length in inches, $w$ is the width in inches, and $h$ is the height in inches. find the length of the tallest scale model that will fit inside deons suitcase. round your answer to the nearest tenth of an inch.
72. balloons the radius of a sphere $r$ is given in terms of its volume $v$ by the formula $r = sqrt3{\frac{0.75v}{pi}}$. by how many inches has the radius of a spherical balloon increased when the amount of air in the balloon is increased from 4.5 cubic feet to 4.7 cubic feet? round your answer to the nearest hundredth

Question

Accepted Answer

# Explanation:
## Step1: Factor radicand into squares
$\sqrt{245m^9n^5} = \sqrt{49 \cdot 5 \cdot m^8 \cdot m \cdot n^4 \cdot n}$
## Step2: Simplify square roots
$\sqrt{49m^8n^4} \cdot \sqrt{5mn} = 7m^4n^2\sqrt{5mn}$

# Answer:
$7m^4n^2\sqrt{5mn}$

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# Explanation:
## Step1: Multiply coefficients and radicands
$2 \cdot 7 \cdot \sqrt{5 \cdot 10}$
## Step2: Simplify product inside root
$14 \cdot \sqrt{50} = 14 \cdot \sqrt{25 \cdot 2}$
## Step3: Simplify square root
$14 \cdot 5\sqrt{2} = 70\sqrt{2}$

# Answer:
$70\sqrt{2}$

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# Explanation:
## Step1: Simplify fraction inside root
$\sqrt{\frac{96d^4e^2f^8}{75de^6}} = \sqrt{\frac{32d^3f^8}{25e^4}}$
## Step2: Split root into numerator/denominator
$\frac{\sqrt{32d^3f^8}}{\sqrt{25e^4}}$
## Step3: Simplify each square root
$\frac{4d f^4\sqrt{2d}}{5e^2}$

# Answer:
$\frac{4d f^4\sqrt{2d}}{5e^2}$

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# Explanation:
## Step1: Factor radicand into cubes
$-11\sqrt[3]{250} = -11\sqrt[3]{125 \cdot 2}$
## Step2: Simplify cube root
$-11 \cdot 5\sqrt[3]{2} = -55\sqrt[3]{2}$

# Answer:
$-55\sqrt[3]{2}$

---

# Explanation:
## Step1: Factor radicand into cubes
$\sqrt[3]{320x^{14}y^{17}z^{20}} = \sqrt[3]{64 \cdot 5 \cdot x^{12} \cdot x^2 \cdot y^{15} \cdot y^2 \cdot z^{18} \cdot z^2}$
## Step2: Simplify cube root
$4x^4y^5z^6\sqrt[3]{5x^2y^2z^2}$

# Answer:
$4x^4y^5z^6\sqrt[3]{5x^2y^2z^2}$

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# Explanation:
## Step1: Multiply radicands
$\sqrt[3]{45k^4m^{10} \cdot 32k^7m^3} = \sqrt[3]{1440k^{11}m^{13}}$
## Step2: Factor into cubes
$\sqrt[3]{216 \cdot \frac{20}{3} \cdot k^9 \cdot k^2 \cdot m^{12} \cdot m} = \sqrt[3]{216k^9m^{12} \cdot 20k^2m}$
## Step3: Simplify cube root
$6k^3m^4\sqrt[3]{20k^2m}$

# Answer:
$6k^3m^4\sqrt[3]{20k^2m}$

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# Explanation:
## Step1: Simplify fraction inside root
$\sqrt[3]{\frac{264w^6xy^7}{3w^6x^4y^3}} = \sqrt[3]{\frac{88y^4}{x^3}}$
## Step2: Split root into numerator/denominator
$\frac{\sqrt[3]{88y^4}}{\sqrt[3]{x^3}}$
## Step3: Simplify cube roots
$\frac{y\sqrt[3]{88y}}{x}$

# Answer:
$\frac{y\sqrt[3]{88y}}{x}$

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# Explanation:
## Step1: Split root into numerator/denominator
$\frac{\sqrt[3]{48a^7}}{\sqrt[3]{125b^9}}$
## Step2: Factor into cubes
$\frac{\sqrt[3]{8 \cdot 6 \cdot a^6 \cdot a}}{5b^3}$
## Step3: Simplify cube root
$\frac{2a^2\sqrt[3]{6a}}{5b^3}$

# Answer:
$\frac{2a^2\sqrt[3]{6a}}{5b^3}$

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# Explanation:
## Step1: Use triangle area formula
$Area = \frac{1}{2} 	imes base 	imes height$
## Step2: Substitute given values
$Area = \frac{1}{2} 	imes 6\sqrt{2} 	imes 3\sqrt{6}$
## Step3: Multiply terms and simplify
$\frac{1}{2} 	imes 18 	imes \sqrt{12} = 9 	imes 2\sqrt{3} = 18\sqrt{3}$

# Answer:
$18\sqrt{3}$ square meters

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# Explanation:
## Step1: Use rectangle area formula
$Area = length 	imes width$
## Step2: Substitute given values
$Area = 8\sqrt{12} 	imes 4\sqrt{8}$
## Step3: Multiply terms and simplify
$32 	imes \sqrt{96} = 32 	imes 4\sqrt{6} = 128\sqrt{6}$

# Answer:
$128\sqrt{6}$ square centimeters

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# Explanation:
## Step1: Substitute values into formula
$	ext{Geometric Mean} = \sqrt{32 	imes 14}$
## Step2: Simplify product inside root
$\sqrt{448} = \sqrt{64 	imes 7}$
## Step3: Simplify square root
$8\sqrt{7}$

# Answer:
$8\sqrt{7}$

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# Explanation:
## Step1: Substitute $A=5$ into formula
$s = \sqrt{\frac{5}{6}}$
## Step2: Calculate decimal value
$\sqrt{\frac{5}{6}} \approx \sqrt{0.8333} \approx 0.9$

# Answer:
$0.9$ yards

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# Explanation:
## Step1: Substitute values into formula
$D = \sqrt{22^2 + 14^2 + 9^2}$
## Step2: Calculate squared terms
$D = \sqrt{484 + 196 + 81}$
## Step3: Sum and simplify root
$D = \sqrt{761} \approx 27.6$

# Answer:
$27.6$ inches

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# Explanation:
## Step1: Calculate radius for $V=4.5$
$r_1 = \sqrt[3]{\frac{0.75 	imes 4.5}{\pi}} = \sqrt[3]{\frac{3.375}{\pi}} \approx \sqrt[3]{1.074} \approx 1.024$ feet
## Step2: Calculate radius for $V=4.7$
$r_2 = \sqrt[3]{\frac{0.75 	imes 4.7}{\pi}} = \sqrt[3]{\frac{3.525}{\pi}} \approx \sqrt[3]{1.122} \approx 1.039$ feet
## Step3: Find difference, convert to inches
$(1.039 - 1.024) 	imes 12 \approx 0.015 	imes 12 = 0.18$ inches

# Answer:
$0.18$ inches

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Factor radicand into squares

Step2: Simplify square roots

Step1: Multiply coefficients and radicands

Step2: Simplify product inside root

Step3: Simplify square root

Step1: Simplify fraction inside root

Step2: Split root into numerator/denominator

Step3: Simplify each square root

Step1: Factor radicand into cubes

Step2: Simplify cube root

Answer: