$$ \left. \begin{array} { l } { 5.14 ( \frac { 5 x ^ { - 1 } } { 2 } ) ^ { - 1 } \cdot \sqrt { 25 x ^ { 4 } y ^ { - 2 } } } \ { 5.15 \sqrt { \frac { ( a b ^ { 2 } ) ^ { 3 } } { a ( - 4 c ) ^ { 2 } } } } \ { 5.16 \frac { ( 3 x ^ { 5 } y ^ { - 3 } ) ^ { 2 } } { - 6 y ^ { 5 } } imes \frac { 2 x ^ { - 4 } } { x ^ { 0 } } } \ { 5.17 \frac { 2 x y ^ { 2 } } { 3 m } \div \frac { 4 y ^ { 2 } } { 9 x m } } \ { 5.18 \frac { 3 x } { y ^ { 2 } z ^ { 3 } } \div \frac { 9 x ^ { 2 } } { 2 y z ^ { 4 } } } \ { 5.19 \frac { ( 2 a c ^ { 2 } ) ^ { 4 } } { 4 a ^ { 4 } } imes \frac { ( a ^ { 2 } b ^ { 3 } ) ^ { 2 } } { ( b ^ { 2 } c ) ^ { 3 } } } \ { 5.20 \frac { ( 2 a ^ { 2 } b ^ { 3 } ) ^ { 4 } imes ( 8 a b ^ { - 2 } ) ^ { 2 } } { 4 a b ^ { 5 } imes 12 a ^ { 9 } b ^ { 7 } } } \ { 5.21 \frac { ( 5 x ^ { - 3 } y ) ^ { 2 } } { 4 x y imes ( x ^ { 2 } y ) ^ { 2 } } \div ( \frac { 5 x y } { 2 } ) ^ { 2 } } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 5.14 ( \frac { 5 x ^ { - 1 } } { 2 } ) ^ { - 1 } \cdot \sqrt { 25 x ^ { 4 } y ^ { - 2 } } } \ { 5.15 \sqrt { \frac { ( a b ^ { 2 } ) ^ { 3 } } { a ( - 4 c ) ^ { 2 } } } } \ { 5.16 \frac { ( 3 x ^ { 5 } y ^ { - 3 } ) ^ { 2 } } { - 6 y ^ { 5 } } 	imes \frac { 2 x ^ { - 4 } } { x ^ { 0 } } } \ { 5.17 \frac { 2 x y ^ { 2 } } { 3 m } \div \frac { 4 y ^ { 2 } } { 9 x m } } \ { 5.18 \frac { 3 x } { y ^ { 2 } z ^ { 3 } } \div \frac { 9 x ^ { 2 } } { 2 y z ^ { 4 } } } \ { 5.19 \frac { ( 2 a c ^ { 2 } ) ^ { 4 } } { 4 a ^ { 4 } } 	imes \frac { ( a ^ { 2 } b ^ { 3 } ) ^ { 2 } } { ( b ^ { 2 } c ) ^ { 3 } } } \ { 5.20 \frac { ( 2 a ^ { 2 } b ^ { 3 } ) ^ { 4 } 	imes ( 8 a b ^ { - 2 } ) ^ { 2 } } { 4 a b ^ { 5 } 	imes 12 a ^ { 9 } b ^ { 7 } } } \ { 5.21 \frac { ( 5 x ^ { - 3 } y ) ^ { 2 } } { 4 x y 	imes ( x ^ { 2 } y ) ^ { 2 } } \div ( \frac { 5 x y } { 2 } ) ^ { 2 } } \end{array} ight. $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\frac{2xy^{2}}{3m}}{\frac{4y^{2}}{9xm}}$$
- step1: Multiply by the reciprocal:
  $$\frac{2xy^{2}}{3m}\times \frac{9xm}{4y^{2}}$$
- step2: Reduce the fraction:
  $$x\times \frac{3x}{2}$$
- step3: Multiply the terms:
  $$\frac{x\times 3x}{2}$$
- step4: Multiply the terms:
  $$\frac{3x^{2}}{2}$$
Calculate or simplify the expression $$ ((2 * a * c^2)^4 / (4 * a^4)) * ((a^2 * b^3)^2 / (b^2 * c)^3) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{\left(2ac^{2}\right)^{4}}{4a^{4}}\right)\left(\frac{\left(a^{2}b^{3}\right)^{2}}{\left(b^{2}c\right)^{3}}\right)$$
- step1: Divide the terms:
  $$4c^{8}\left(\frac{\left(a^{2}b^{3}\right)^{2}}{\left(b^{2}c\right)^{3}}\right)$$
- step2: Remove the parentheses:
  $$4c^{8}\times \frac{\left(a^{2}b^{3}\right)^{2}}{\left(b^{2}c\right)^{3}}$$
- step3: Rewrite the expression:
  $$4c^{8}\times \frac{\left(a^{2}b^{3}\right)^{2}}{c^{3}b^{6}}$$
- step4: Reduce the fraction:
  $$4c^{5}\times \frac{\left(a^{2}b^{3}\right)^{2}}{b^{6}}$$
- step5: Multiply the terms:
  $$\frac{4c^{5}\left(a^{2}b^{3}\right)^{2}}{b^{6}}$$
- step6: Multiply the terms:
  $$\frac{4c^{5}a^{4}b^{6}}{b^{6}}$$
- step7: Reduce the fraction:
  $$4c^{5}a^{4}$$
Calculate or simplify the expression $$ (3 * x / (y^2 * z^3)) / (9 * x^2 / (2 * y * z^4)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\frac{3x}{y^{2}z^{3}}}{\frac{9x^{2}}{2yz^{4}}}$$
- step1: Multiply by the reciprocal:
  $$\frac{3x}{y^{2}z^{3}}\times \frac{2yz^{4}}{9x^{2}}$$
- step2: Reduce the fraction:
  $$\frac{1}{y}\times \frac{2z}{3x}$$
- step3: Multiply the terms:
  $$\frac{2z}{y\times 3x}$$
- step4: Multiply the terms:
  $$\frac{2z}{3yx}$$
Calculate or simplify the expression $$ ((5 * x^(-3) * y)^2 / (4 * x * y * (x^2 * y)^2)) / ((5 * x * y / 2)^2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\frac{\left(5x^{-3}y\right)^{2}}{\left(4xy\left(x^{2}y\right)^{2}\right)}\right)}{\left(\left(\frac{5xy}{2}\right)^{2}\right)}$$
- step1: Evaluate:
  $$\frac{\left(\frac{\left(5x^{-3}y\right)^{2}}{\left(4xy\left(x^{2}y\right)^{2}\right)}\right)}{\left(\frac{5xy}{2}\right)^{2}}$$
- step2: Remove the parentheses:
  $$\frac{\frac{\left(5x^{-3}y\right)^{2}}{4xy\left(x^{2}y\right)^{2}}}{\left(\frac{5xy}{2}\right)^{2}}$$
- step3: Multiply the terms:
  $$\frac{\frac{\left(5x^{-3}y\right)^{2}}{4x^{5}y^{3}}}{\left(\frac{5xy}{2}\right)^{2}}$$
- step4: Rewrite the expression:
  $$\frac{\frac{\left(5x^{-3}y\right)^{2}}{4x^{5}y^{3}}}{\frac{25x^{2}y^{2}}{4}}$$
- step5: Multiply by the reciprocal:
  $$\frac{\left(5x^{-3}y\right)^{2}}{4x^{5}y^{3}}\times \frac{4}{25x^{2}y^{2}}$$
- step6: Rewrite the expression:
  $$\frac{25x^{-6}y^{2}}{4x^{5}y^{3}}\times \frac{4}{25x^{2}y^{2}}$$
- step7: Reduce the fraction:
  $$\frac{x^{-6}}{x^{5}y^{3}}\times \frac{1}{x^{2}}$$
- step8: Multiply the terms:
  $$\frac{x^{-6}}{x^{5}y^{3}x^{2}}$$
- step9: Multiply the terms:
  $$\frac{x^{-6}}{x^{7}y^{3}}$$
- step10: Reduce the fraction:
  $$\frac{1}{x^{13}y^{3}}$$
Calculate or simplify the expression $$ ((2 * a^2 * b^3)^4 * (8 * a * b^(-2))^2) / (4 * a * b^5 * 12 * a^9 * b^7) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\left(2a^{2}b^{3}\right)^{4}\left(8ab^{-2}\right)^{2}\right)}{\left(4ab^{5}\times 12a^{9}b^{7}\right)}$$
- step1: Remove the parentheses:
  $$\frac{\left(2a^{2}b^{3}\right)^{4}\left(8ab^{-2}\right)^{2}}{4ab^{5}\times 12a^{9}b^{7}}$$
- step2: Multiply:
  $$\frac{\left(2a^{2}b^{3}\right)^{4}\left(8ab^{-2}\right)^{2}}{48a^{10}b^{12}}$$
- step3: Factor the expression:
  $$\frac{16a^{8}b^{12}\left(8ab^{-2}\right)^{2}}{48a^{10}b^{12}}$$
- step4: Reduce the fraction:
  $$\frac{\left(8ab^{-2}\right)^{2}}{3a^{2}}$$
- step5: Evaluate the power:
  $$\frac{64a^{2}b^{-4}}{3a^{2}}$$
- step6: Reduce the fraction:
  $$\frac{64b^{-4}}{3}$$
- step7: Rewrite the expression:
  $$\frac{\frac{64}{b^{4}}}{3}$$
- step8: Simplify:
  $$\frac{64}{3b^{4}}$$
Calculate or simplify the expression $$ \sqrt((a * b^2)^3 / (a * (-4 * c)^2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\sqrt{\frac{\left(ab^{2}\right)^{3}}{\left(a\left(-4c\right)^{2}\right)}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{\left(ab^{2}\right)^{3}}{a\left(-4c\right)^{2}}}$$
- step2: Reduce the fraction:
  $$\sqrt{\frac{a^{2}b^{6}}{16c^{2}}}$$
- step3: Use the properties of radicals:
  $$\frac{\sqrt{a^{2}b^{6}}}{\sqrt{16c^{2}}}$$
- step4: Simplify the expression:
  $$\frac{ab^{3}}{4c}$$
Calculate or simplify the expression $$ 5.14 * (5 * x^(-1) / 2)^(-1) * \sqrt(25 * x^4 * y^(-2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$5.14\left(\frac{5x^{-1}}{2}\right)^{-1}\sqrt{25x^{4}y^{-2}}$$
- step1: Rewrite the expression:
  $$5.14\left(\frac{5}{2x}\right)^{-1}\sqrt{25x^{4}y^{-2}}$$
- step2: Simplify the root:
  $$5.14\left(\frac{5}{2x}\right)^{-1}\times \frac{5x^{2}}{y}$$
- step3: Multiply the terms:
  $$\frac{257x}{125}\times \frac{5x^{2}}{y}$$
- step4: Reduce the fraction:
  $$\frac{257x}{25}\times \frac{x^{2}}{y}$$
- step5: Multiply the terms:
  $$\frac{257x\times x^{2}}{25y}$$
- step6: Multiply the terms:
  $$\frac{257x^{3}}{25y}$$
Calculate or simplify the expression $$ ((3 * x^5 * y^(-3))^2 / (-6 * y^5)) * (2 * x^(-4) / x^0) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{\left(3x^{5}y^{-3}\right)^{2}}{\left(-6y^{5}\right)}\right)\left(\frac{2x^{-4}}{x^{0}}\right)$$
- step1: Remove the parentheses:
  $$\left(\frac{\left(3x^{5}y^{-3}\right)^{2}}{-6y^{5}}\right)\left(\frac{2x^{-4}}{x^{0}}\right)$$
- step2: Multiply by $$a^{-n}:$$
  $$\frac{\left(3x^{5}y^{-3}\right)^{2}}{-6y^{5}}\times 2x^{-4}\times x^{0}$$
- step3: Divide the terms:
  $$-\frac{3^{2}x^{10}}{6y^{11}}\times 2x^{-4}\times x^{0}$$
- step4: Evaluate the power:
  $$-\frac{3^{2}x^{10}}{6y^{11}}\times 2x^{-4}\times 1$$
- step5: Rewrite the expression:
  $$-\frac{3^{2}x^{10}}{6y^{11}}\times 2x^{-4}$$
- step6: Multiply the terms:
  $$-\frac{3x^{6}}{y^{11}}$$
Let's solve each of the problems step by step:

### Problem 5.14
$$
5.14 \left( \frac{5 x^{-1}}{2} \right)^{-1} \cdot \sqrt{25 x^{4} y^{-2}}
$$

1. **Calculate the first part:**
   $$
   \left( \frac{5 x^{-1}}{2} \right)^{-1} = \frac{2}{5} x
   $$

2. **Calculate the square root:**
   $$
   \sqrt{25 x^{4} y^{-2}} = 5 x^{2} y^{-1} = \frac{5 x^{2}}{y}
   $$

3. **Combine the results:**
   $$
   5.14 \cdot \frac{2}{5} x \cdot \frac{5 x^{2}}{y} = \frac{257 x^{3}}{25 y}
   $$

### Problem 5.15
$$
\sqrt{\frac{(a b^{2})^{3}}{a (-4 c)^{2}}}
$$

1. **Calculate the numerator:**
   $$
   (a b^{2})^{3} = a^{3} b^{6}
   $$

2. **Calculate the denominator:**
   $$
   a (-4 c)^{2} = a \cdot 16 c^{2} = 16 a c^{2}
   $$

3. **Combine and simplify:**
   $$
   \sqrt{\frac{a^{3} b^{6}}{16 a c^{2}}} = \sqrt{\frac{ab^{6}}{16c^{2}}} = \frac{b^{3}}{4c}
   $$

### Problem 5.16
$$
\frac{(3 x^{5} y^{-3})^{2}}{-6 y^{5}} \times \frac{2 x^{-4}}{x^{0}}
$$

1. **Calculate the first part:**
   $$
   (3 x^{5} y^{-3})^{2} = 9 x^{10} y^{-6}
   $$

2. **Combine with the second part:**
   $$
   \frac{9 x^{10} y^{-6}}{-6 y^{5}} = -\frac{3 x^{10}}{2 y^{11}}
   $$

3. **Multiply by the second fraction:**
   $$
   -\frac{3 x^{10}}{2 y^{11}} \times 2 x^{-4} = -\frac{3 x^{6}}{y^{11}} = -\frac{3 x^{6}}{y^{11}}
   $$

### Problem 5.17
$$
\frac{2 x y^{2}}{3 m} \div \frac{4 y^{2}}{9 x m}
$$

1. **Rewrite as multiplication:**
   $$
   \frac{2 x y^{2}}{3 m} \times \frac{9 x m}{4 y^{2}} = \frac{18 x^{2}}{12} = \frac{3 x^{2}}{2}
   $$

### Problem 5.18
$$
\frac{3 x}{y^{2} z^{3}} \div \frac{9 x^{2}}{2 y z^{4}}
$$

1. **Rewrite as multiplication:**
   $$
   \frac{3 x}{y^{2} z^{3}} \times \frac{2 y z^{4}}{9 x^{2}} = \frac{6 z}{9 y^{2} x} = \frac{2 z}{3 y x}
   $$

### Problem 5.19
$$
\frac{(2 a c^{2})^{4}}{4 a^{4}} \times \frac{(a^{2} b^{3})^{2}}{(b^{2} c)^{3}}
$$

1. **Calculate the first part:**
   $$
   (2 a c^{2})^{4} = 16 a^{4} c^{8}
   $$
   $$
   \frac{16 a^{4} c^{8}}{4 a^{4}} = 4 c^{8}
   $$

2. **Calculate the second part:**
   $$
   (a^{2} b^{3})^{2} = a^{4} b^{6}
   $$
   $$
   (b^{2} c)^{3} = b^{6} c^{3}
   $$
   $$
   \frac{a^{4} b^{6}}{b^{6} c^{3}} = \frac{a^{4}}{c^{3}}
   $$

3. **Combine:**
   $$
   4 c^{8} \cdot \frac{a^{4}}{c^{3}} = 4 a^{4} c^{5}
   $$

### Problem 5.20
$$
\frac{(2 a^{2} b^{3})^{4} \times (8 a b^{-2})^{2}}{4 a b^{5} \times 12 a^{9} b^{7}}
$$

1. **Calculate the numerator:**
   $$
   (2 a^{2} b^{3})^{4} = 16 a^{8} b^{12}
   $$
   $$
   (8 a b^{-2})^{2} = 64 a^{2} b^{-4}
   $$
   $$
   16 a^{8} b^{12} \times 64 a^{2} b^{-4} = 1024 a^{10} b^{8}
   $$

2. **Calculate the denominator:**
   $$
   4 a b^{5} \times 12 a^{9} b^{7} = 48 a^{10} b^{12}
   $$

3. **Combine:**
   $$
   \frac{1024 a^{10} b^{8}}{48 a^{10} b^{12}} = \frac{1024}{48 b^{4}} = \frac{64}{3 b^{4}}
   $$

### Problem 5.21
$$
\frac{(5 x^{-3} y)^{2}}{4 x y \times (x^{2} y)^{2}} \div \left( \frac{5 x y}{2} \right)^{2}
$$

1. **Calculate the numerator:**
   $$
   (5 x^{-3} y)^{2} = 25 x^{-6} y^{2}
   $$

2. **Calculate the denominator:**
   $$
   4 x y \times (x^{2} y)^{2} = 4 x y \times x^{4} y^{
Here are the simplified solutions for each problem: 1. **5.14** \[ \frac{257 x^{3}}{25 y} $$ 2. **5.15** $$ \frac{b^{3}}{4c} $$ 3. **5.16** $$ -\frac{3 x^{6}}{y^{11}} $$ 4. **5.17** $$ \frac{3 x^{2}}{2} $$ 5. **5.18** $$ \frac{2 z}{3 y x} $$ 6. **5.19** $$ 4 a^{4} c^{5} $$ 7. **5.20** $$ \frac{64}{3 b^{4}} $$ 8. **5.21** $$ \frac{25 x^{-6} y^{2}}{4 x y \times x^{4} y^{2}} \div \left( \frac{5 x y}{2} \right)^{2} $$ (Further simplification required for this problem.)

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