Question

1. $(3u^{-4})^{-5}$
2. $2a^{-9} \cdot 8a^{6}$
3. $-6n^{10} \cdot 6n^{-3}$
4. $6x^{10} \cdot -9x^{5}$
5. $9m^{-8} \cdot 6m^{-6}$
6. $\frac{5u^{8}v^{-10}}{30uv^{6}}$
7. $\frac{6x^{5}y^{-16}}{3x^{8}y^{-10}}$
8. $\frac{2m^{5}n^{-8}}{-10m^{-9}n^{10}}$
9. $\frac{3x^{-2}y^{-7}}{-6x^{-8}y^{-2}}$
10. $\frac{90x^{10}y^{4}}{35x^{-10}y^{-1}}$

Explanation:

11) Step1: Apply power of a power rule

$(3u^{-4})^{-5} = 3^{-5} \cdot u^{(-4)\times(-5)}$

11) Step2: Simplify exponents and constants

$3^{-5} = \frac{1}{3^5} = \frac{1}{243}$, $u^{20}$
$\frac{1}{243}u^{20} = \frac{u^{20}}{243}$

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12) Step1: Multiply coefficients first

$2a^{-9} \cdot 8a^6 = (2 \times 8) \cdot a^{-9+6}$

12) Step2: Simplify coefficients and exponents

$16 \cdot a^{-3} = \frac{16}{a^3}$

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13) Step1: Multiply coefficients and exponents

$-6n^{10} \cdot 6n^{-3} = (-6 \times 6) \cdot n^{10+(-3)}$

13) Step2: Simplify the result

$-36 \cdot n^{7} = -36n^7$

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14) Step1: Multiply coefficients and exponents

$6x^{10} \cdot -9x^5 = (6 \times -9) \cdot x^{10+5}$

14) Step2: Simplify the result

$-54 \cdot x^{15} = -54x^{15}$

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15) Step1: Multiply coefficients and exponents

$9m^{-8} \cdot 6m^{-6} = (9 \times 6) \cdot m^{-8+(-6)}$

15) Step2: Simplify the result

$54 \cdot m^{-14} = \frac{54}{m^{14}}$

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16) Step1: Simplify coefficients and $u$ terms

$\frac{5u^8v^{-10}}{30uv^6} = \frac{5}{30} \cdot \frac{u^8}{u} \cdot \frac{v^{-10}}{v^6}$

16) Step2: Simplify each fraction

$\frac{1}{6} \cdot u^{8-1} \cdot v^{-10-6} = \frac{1}{6}u^7v^{-16} = \frac{u^7}{6v^{16}}$

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17) Step1: Simplify coefficients and variables

$\frac{6x^5y^{-16}}{3x^8y^{-10}} = \frac{6}{3} \cdot \frac{x^5}{x^8} \cdot \frac{y^{-16}}{y^{-10}}$

17) Step2: Simplify each fraction

$2 \cdot x^{5-8} \cdot y^{-16-(-10)} = 2x^{-3}y^{-6} = \frac{2}{x^3y^6}$

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18) Step1: Simplify coefficients and variables

$\frac{2m^5n^{-8}}{-10m^{-9}n^{10}} = \frac{2}{-10} \cdot \frac{m^5}{m^{-9}} \cdot \frac{n^{-8}}{n^{10}}$

18) Step2: Simplify each fraction

$-\frac{1}{5} \cdot m^{5-(-9)} \cdot n^{-8-10} = -\frac{1}{5}m^{14}n^{-18} = -\frac{m^{14}}{5n^{18}}$

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19) Step1: Simplify coefficients and variables

$\frac{3x^{-2}y^{-7}}{-6x^{-8}y^{-2}} = \frac{3}{-6} \cdot \frac{x^{-2}}{x^{-8}} \cdot \frac{y^{-7}}{y^{-2}}$

19) Step2: Simplify each fraction

$-\frac{1}{2} \cdot x^{-2-(-8)} \cdot y^{-7-(-2)} = -\frac{1}{2}x^{6}y^{-5} = -\frac{x^6}{2y^5}$

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20) Step1: Simplify coefficients and variables

$\frac{90x^{10}y^4}{35x^{-10}y^{-1}} = \frac{90}{35} \cdot \frac{x^{10}}{x^{-10}} \cdot \frac{y^4}{y^{-1}}$

20) Step2: Simplify each fraction

$\frac{18}{7} \cdot x^{10-(-10)} \cdot y^{4-(-1)} = \frac{18}{7}x^{20}y^{5} = \frac{18x^{20}y^5}{7}$

Answer:

1. $\frac{u^{20}}{243}$
2. $\frac{16}{a^3}$
3. $-36n^7$
4. $-54x^{15}$
5. $\frac{54}{m^{14}}$
6. $\frac{u^7}{6v^{16}}$
7. $\frac{2}{x^3y^6}$
8. $-\frac{m^{14}}{5n^{18}}$
9. $-\frac{x^6}{2y^5}$
10. $\frac{18x^{20}y^5}{7}$