5. Simplifica las siguientes expresiones. $$ \begin{array}{ll} ext { a. } \frac{25^{3} \cdot\left(10^{3} ight)^{4} \cdot 169}{13 \cdot 125 \cdot 10^{9}} & ext { d. } \frac{10^{25} \cdot 495}{\left(210^{4} ight)^{2} \cdot 100^{5} \%} \ ext { b. } \frac{\left[(15)^{2} ight]^{2} \cdot 81 \cdot 8}{108 \cdot 9^{-4} \cdot 225} & ext { e. }\left(-7 x^{4} y^{3} ight)^{2} \cdot\left(5 x^{2} 4 ight) \ ext { c. } \frac{30^{20} \cdot 15^{2}}{6^{14}} & ext { f. }\left(\frac{6 x^{4} y^{-2}}{12 x y^{3} z^{-1}} ight)^{-3}\end{array} $$

Question

5. Simplifica las siguientes expresiones. $$ \begin{array}{ll}	ext { a. } \frac{25^{3} \cdot\left(10^{3}ight)^{4} \cdot 169}{13 \cdot 125 \cdot 10^{9}} & 	ext { d. } \frac{10^{25} \cdot 495}{\left(210^{4}ight)^{2} \cdot 100^{5} \%} \ 	ext { b. } \frac{\left[(15)^{2}ight]^{2} \cdot 81 \cdot 8}{108 \cdot 9^{-4} \cdot 225} & 	ext { e. }\left(-7 x^{4} y^{3}ight)^{2} \cdot\left(5 x^{2} 4ight) \ 	ext { c. } \frac{30^{20} \cdot 15^{2}}{6^{14}} & 	ext { f. }\left(\frac{6 x^{4} y^{-2}}{12 x y^{3} z^{-1}}ight)^{-3}\end{array} $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(30^{20}\times 15^{2}\right)}{6^{14}}$$
- step1: Remove the parentheses:
  $$\frac{30^{20}\times 15^{2}}{6^{14}}$$
- step2: Factor the expression:
  $$\frac{6^{20}\times 5^{20}\times 15^{2}}{6^{14}}$$
- step3: Reduce the fraction:
  $$6^{6}\times 5^{20}\times 15^{2}$$
- step4: Expand the expression:
  $$46656\times 5^{20}\times 15^{2}$$
- step5: Multiply the terms:
  $$10497600\times 5^{20}$$
Calculate or simplify the expression $$ (10^25 * 495) / ((210^4)^2 * 100^5) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(10^{25}\times 495\right)}{\left(\left(210^{4}\right)^{2}\times 100^{5}\right)}$$
- step1: Remove the parentheses:
  $$\frac{10^{25}\times 495}{\left(210^{4}\right)^{2}\times 100^{5}}$$
- step2: Multiply the exponents:
  $$\frac{10^{25}\times 495}{210^{4\times 2}\times 100^{5}}$$
- step3: Multiply the numbers:
  $$\frac{10^{25}\times 495}{210^{8}\times 100^{5}}$$
- step4: Factor the expression:
  $$\frac{10^{25}\times 495}{10^{8}\times 21^{8}\times 100^{5}}$$
- step5: Reduce the fraction:
  $$\frac{10^{17}\times 495}{21^{8}\times 100^{5}}$$
- step6: Factor the expression:
  $$\frac{10^{17}\times 495}{21^{8}\times 10^{10}}$$
- step7: Reduce the fraction:
  $$\frac{10^{7}\times 495}{21^{8}}$$
- step8: Factor the expression:
  $$\frac{10^{7}\times 9\times 55}{21^{8}}$$
- step9: Factor the expression:
  $$\frac{10^{7}\times 9\times 55}{3^{8}\times 7^{8}}$$
- step10: Reduce the fraction:
  $$\frac{10^{7}\times 55}{729\times 7^{8}}$$
- step11: Calculate:
  $$\frac{55\times 10^{7}}{729\times 7^{8}}$$
Calculate or simplify the expression $$ ((6 * x^4 * y^(-2)) / (12 * x * y^3 * z^(-1)))^(-3) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{\left(6x^{4}y^{-2}\right)}{\left(12xy^{3}z^{-1}\right)}\right)^{-3}$$
- step1: Remove the parentheses:
  $$\left(\frac{6x^{4}y^{-2}}{12xy^{3}z^{-1}}\right)^{-3}$$
- step2: Multiply by $$a^{-n}:$$
  $$\left(\frac{6x^{4}y^{-2}x^{-1}y^{-3}}{12z^{-1}}\right)^{-3}$$
- step3: Reduce the fraction:
  $$\left(\frac{x^{3}y^{-5}}{2z^{-1}}\right)^{-3}$$
- step4: Rewrite the expression:
  $$\left(\frac{x^{3}z}{2y^{5}}\right)^{-3}$$
- step5: Transform the expression:
  $$\left(\frac{2y^{5}}{x^{3}z}\right)^{3}$$
- step6: Rewrite the expression:
  $$\frac{\left(2y^{5}\right)^{3}}{\left(x^{3}z\right)^{3}}$$
- step7: Evaluate the power:
  $$\frac{8y^{15}}{x^{9}z^{3}}$$
Calculate or simplify the expression $$ ((-7 * x^4 * y^3)^2 * (5 * x^2 * 4)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(-7x^{4}y^{3}\right)^{2}\left(5x^{2}\times 4\right)$$
- step1: Remove the parentheses:
  $$\left(-7x^{4}y^{3}\right)^{2}\times 5x^{2}\times 4$$
- step2: Multiply the terms:
  $$\left(-7x^{4}y^{3}\right)^{2}\times 20x^{2}$$
- step3: Multiply the terms:
  $$980x^{8}y^{6}x^{2}$$
- step4: Multiply the terms:
  $$980x^{10}y^{6}$$
Calculate or simplify the expression $$ (25^3 * (10^3)^4 * 169) / (13 * 125 * 10^9) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(25^{3}\left(10^{3}\right)^{4}\times 169\right)}{\left(13\times 125\times 10^{9}\right)}$$
- step1: Remove the parentheses:
  $$\frac{25^{3}\left(10^{3}\right)^{4}\times 169}{13\times 125\times 10^{9}}$$
- step2: Multiply the exponents:
  $$\frac{25^{3}\times 10^{3\times 4}\times 169}{13\times 125\times 10^{9}}$$
- step3: Multiply by $$a^{-n}:$$
  $$\frac{25^{3}\times 10^{3\times 4}\times 169\times 10^{-9}}{13\times 125}$$
- step4: Multiply the numbers:
  $$\frac{25^{3}\times 10^{12}\times 169\times 10^{-9}}{13\times 125}$$
- step5: Multiply:
  $$\frac{25^{3}\times 10^{3}\times 169}{13\times 125}$$
- step6: Multiply the numbers:
  $$\frac{25^{3}\times 10^{3}\times 169}{1625}$$
- step7: Reduce the fraction:
  $$\frac{25^{3}\times 10^{3}\times 13}{125}$$
- step8: Factor the expression:
  $$\frac{5^{6}\times 10^{3}\times 13}{5^{3}}$$
- step9: Reduce the fraction:
  $$5^{3}\times 10^{3}\times 13$$
- step10: Multiply the numbers:
  $$50^{3}\times 13$$
- step11: Multiply the terms:
  $$13\times 50^{3}$$
Calculate or simplify the expression $$ ([(15)^2]^2 * 81 * 8) / (108 * 9^(-4) * 225) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(\left(15^{2}\right)^{2}\times 81\times 8\right)}{\left(108\times 9^{-4}\times 225\right)}$$
- step1: Remove the parentheses:
  $$\frac{\left(15^{2}\right)^{2}\times 81\times 8}{108\times 9^{-4}\times 225}$$
- step2: Multiply the exponents:
  $$\frac{15^{2\times 2}\times 81\times 8}{108\times 9^{-4}\times 225}$$
- step3: Multiply the numbers:
  $$\frac{15^{4}\times 81\times 8}{108\times 9^{-4}\times 225}$$
- step4: Multiply the terms:
  $$\frac{15^{4}\times 648}{108\times 9^{-4}\times 225}$$
- step5: Multiply the terms:
  $$\frac{15^{4}\times 648}{24300\times 9^{-4}}$$
- step6: Reduce the fraction:
  $$\frac{15^{4}\times 2}{75\times 9^{-4}}$$
- step7: Factor the expression:
  $$\frac{15^{4}\times 2}{15\times 5\times 9^{-4}}$$
- step8: Reduce the fraction:
  $$\frac{15^{3}\times 2}{5\times 9^{-4}}$$
- step9: Factor the expression:
  $$\frac{5^{3}\times 3^{3}\times 2}{5\times 9^{-4}}$$
- step10: Reduce the fraction:
  $$\frac{5^{2}\times 3^{3}\times 2}{9^{-4}}$$
- step11: Factor the expression:
  $$\frac{5^{2}\times 3^{3}\times 2}{3^{-8}}$$
- step12: Reduce the fraction:
  $$5^{2}\times 3^{11}\times 2$$
- step13: Expand the expression:
  $$25\times 3^{11}\times 2$$
- step14: Multiply the terms:
  $$50\times 3^{11}$$
Aquí están las simplificaciones de las expresiones solicitadas:

### a. 
$$
\frac{25^{3} \cdot (10^{3})^{4} \cdot 169}{13 \cdot 125 \cdot 10^{9}} = 13 \times 50^{3} = 1625000
$$

### b. 
$$
\frac{[(15)^{2}]^{2} \cdot 81 \cdot 8}{108 \cdot 9^{-4} \cdot 225} = 50 \times 3^{11} = 8857350
$$

### c. 
$$
\frac{30^{20} \cdot 15^{2}}{6^{14}} = 10497600 \times 5^{20} \approx 1.001129 \times 10^{21}
$$

### d. 
$$
\frac{10^{25} \cdot 495}{(210^{4})^{2} \cdot 100^{5}} = \frac{55 \times 10^{7}}{729 \times 7^{8}}
$$

### e. 
$$
(-7 x^{4} y^{3})^{2} \cdot (5 x^{2} \cdot 4) = 980 x^{10} y^{6}
$$

### f. 
$$
\left(\frac{6 x^{4} y^{-2}}{12 x y^{3} z^{-1}}\right)^{-3} = \frac{8 y^{15}}{x^{9} z^{3}}
$$

Si necesitas más detalles sobre algún paso específico, no dudes en preguntar.
a. $$ 1625000 $$ b. $$ 8857350 $$ c. $$ 1.001129 \times 10^{21} $$ d. $$ \frac{55 \times 10^{7}}{729 \times 7^{8}} $$ e. $$ 980 x^{10} y^{6} $$ f. $$ \frac{8 y^{15}}{x^{9} z^{3}} $$

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