Resuelve las siguientes operaciones. a. $$ 2^{-3} \cdot\left(4^{-2} ight)^{4} $$ $$ \begin{array}{ll} ext { b. }\left(\frac{2}{5} ight)^{4} \div\left(\frac{4}{25} ight)^{-2} & ext { e. } \frac{4^{-1}+5^{-1}}{20^{-1}} \ ext { c. }\left(4^{-3} ight)\left[\left(\frac{1}{16} ight)^{-2} ight]^{2} & ext { f. }\left(\frac{3}{4} ight)^{2} \cdot\left(\frac{9}{16} ight)^{-2} \ ext { d. }\left(\frac{5}{3} ight)^{5} \cdot\left(\frac{3}{5} ight)^{5}(3)^{9} & ext { g. } 3^{5} \cdot\left[\left(\frac{1}{2} ight)^{2} ight]^{-3} \ ext { h. }\left(2^{-5} \cdot \frac{1}{4} \cdot \frac{1}{32} ight)\end{array} $$

Question

Resuelve las siguientes operaciones. a. $$ 2^{-3} \cdot\left(4^{-2}ight)^{4} $$ $$ \begin{array}{ll}	ext { b. }\left(\frac{2}{5}ight)^{4} \div\left(\frac{4}{25}ight)^{-2} & 	ext { e. } \frac{4^{-1}+5^{-1}}{20^{-1}} \ 	ext { c. }\left(4^{-3}ight)\left[\left(\frac{1}{16}ight)^{-2}ight]^{2} & 	ext { f. }\left(\frac{3}{4}ight)^{2} \cdot\left(\frac{9}{16}ight)^{-2} \ 	ext { d. }\left(\frac{5}{3}ight)^{5} \cdot\left(\frac{3}{5}ight)^{5}(3)^{9} & 	ext { g. } 3^{5} \cdot\left[\left(\frac{1}{2}ight)^{2}ight]^{-3} \ 	ext { h. }\left(2^{-5} \cdot \frac{1}{4} \cdot \frac{1}{32}ight)\end{array} $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{5}{3}\right)^{5}\left(\frac{3}{5}\right)^{5}\times 3^{9}$$
- step1: Transform the expression:
  $$\left(\frac{5}{3}\right)^{5}\left(\frac{5}{3}\right)^{-5}\times 3^{9}$$
- step2: Multiply the terms:
  $$\left(\frac{5}{3}\right)^{5-5}\times 3^{9}$$
- step3: Subtract the numbers:
  $$\left(\frac{5}{3}\right)^{0}\times 3^{9}$$
- step4: Evaluate the power:
  $$3^{9}$$
- step5: Evaluate the power:
  $$19683$$
Calculate or simplify the expression $$ (4^{-1} + 5^{-1}) / (20^{-1}) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(4^{-1}+5^{-1}\right)}{\left(20^{-1}\right)}$$
- step1: Evaluate:
  $$\frac{\left(4^{-1}+5^{-1}\right)}{20^{-1}}$$
- step2: Remove the parentheses:
  $$\frac{4^{-1}+5^{-1}}{20^{-1}}$$
- step3: Add the numbers:
  $$\frac{\frac{9}{20}}{20^{-1}}$$
- step4: Express with a positive exponent:
  $$\frac{\frac{9}{20}}{\frac{1}{20}}$$
- step5: Multiply by the reciprocal:
  $$\frac{9}{20}\times 20$$
- step6: Reduce the numbers:
  $$9\times 1$$
- step7: Simplify:
  $$9$$
Calculate or simplify the expression $$ (2/5)^{4} / (4/25)^{-2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(\frac{2}{5}\right)^{4}}{\left(\frac{4}{25}\right)^{-2}}$$
- step1: Multiply by the reciprocal:
  $$\left(\frac{2}{5}\right)^{4}\times \frac{4^{2}}{25^{2}}$$
- step2: Evaluate the power:
  $$\frac{2^{4}}{5^{4}}\times \frac{4^{2}}{25^{2}}$$
- step3: Multiply the fractions:
  $$\frac{2^{4}\times 4^{2}}{5^{4}\times 25^{2}}$$
- step4: Multiply:
  $$\frac{2^{8}}{5^{8}}$$
- step5: Evaluate the power:
  $$\frac{256}{5^{8}}$$
Calculate or simplify the expression $$ (4^{-3}) * ((1/16)^{-2})^{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(4^{-3}\right)\left(\left(\frac{1}{16}\right)^{-2}\right)^{2}$$
- step1: Evaluate:
  $$4^{-3}\left(\left(\frac{1}{16}\right)^{-2}\right)^{2}$$
- step2: Multiply the exponents:
  $$4^{-3}\left(\frac{1}{16}\right)^{-2\times 2}$$
- step3: Multiply the numbers:
  $$4^{-3}\left(\frac{1}{16}\right)^{-4}$$
- step4: Rewrite the expression:
  $$2^{-6}\times 2^{16}$$
- step5: Multiply the terms:
  $$2^{10}$$
- step6: Evaluate the power:
  $$1024$$
Calculate or simplify the expression $$ 2^{-3} * (4^{-2})^{4} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2^{-3}\left(4^{-2}\right)^{4}$$
- step1: Multiply the exponents:
  $$2^{-3}\times 4^{-2\times 4}$$
- step2: Multiply the numbers:
  $$2^{-3}\times 4^{-8}$$
- step3: Rewrite the expression:
  $$2^{-3}\times 2^{-16}$$
- step4: Simplify:
  $$2^{-19}$$
- step5: Express with a positive exponent:
  $$\frac{1}{2^{19}}$$
Calculate or simplify the expression $$ 3^{5} * ((1/2)^{2})^{-3} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3^{5}\left(\left(\frac{1}{2}\right)^{2}\right)^{-3}$$
- step1: Multiply the exponents:
  $$3^{5}\left(\frac{1}{2}\right)^{2\left(-3\right)}$$
- step2: Multiply the numbers:
  $$3^{5}\left(\frac{1}{2}\right)^{-6}$$
- step3: Rewrite the expression:
  $$3^{5}\times 2^{6}$$
- step4: Expand the expression:
  $$243\times 2^{6}$$
- step5: Expand the expression:
  $$243\times 64$$
- step6: Multiply the numbers:
  $$15552$$
Calculate or simplify the expression $$ (3/4)^{2} * (9/16)^{-2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{3}{4}\right)^{2}\left(\frac{9}{16}\right)^{-2}$$
- step1: Rewrite the expression:
  $$\left(\frac{3}{4}\right)^{2}\left(\frac{3}{4}\right)^{-4}$$
- step2: Multiply the terms:
  $$\left(\frac{3}{4}\right)^{-2}$$
- step3: Simplify:
  $$\frac{4^{2}}{3^{2}}$$
- step4: Evaluate the power:
  $$\frac{16}{3^{2}}$$
- step5: Evaluate the power:
  $$\frac{16}{9}$$
Calculate or simplify the expression $$ (2^{-5} * (1/4) * (1/32)) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(2^{-5}\left(\frac{1}{4}\right)\left(\frac{1}{32}\right)\right)$$
- step1: Evaluate:
  $$2^{-5}\times \frac{1}{4}\times \frac{1}{32}$$
- step2: Multiply the terms:
  $$2^{-5}\times \frac{1}{128}$$
- step3: Rewrite the expression:
  $$\frac{1}{2^{5}}\times \frac{1}{128}$$
- step4: Multiply the fractions:
  $$\frac{1}{2^{5}\times 128}$$
- step5: Multiply:
  $$\frac{1}{2^{12}}$$
- step6: Evaluate the power:
  $$\frac{1}{4096}$$
Aquí están las soluciones a las operaciones solicitadas:

a. $$ 2^{-3} \cdot (4^{-2})^{4} $$

Primero, simplificamos $$ (4^{-2})^{4} $$:
$$
(4^{-2})^{4} = 4^{-8} = (2^{2})^{-8} = 2^{-16}
$$
Ahora, multiplicamos:
$$
2^{-3} \cdot 2^{-16} = 2^{-3-16} = 2^{-19}
$$
Por lo tanto, el resultado es:
$$
\frac{1}{2^{19}} \quad \text{o aproximadamente } 1.907349 \times 10^{-6}
$$

b. $$ \left(\frac{2}{5}\right)^{4} \div \left(\frac{4}{25}\right)^{-2} $$

Primero, simplificamos $$ \left(\frac{4}{25}\right)^{-2} $$:
$$
\left(\frac{4}{25}\right)^{-2} = \left(\frac{25}{4}\right)^{2} = \frac{625}{16}
$$
Ahora, realizamos la división:
$$
\left(\frac{2}{5}\right)^{4} = \frac{16}{625}
$$
Por lo tanto:
$$
\frac{16}{625} \div \frac{625}{16} = \frac{16}{625} \cdot \frac{16}{625} = \frac{256}{5^{8}} \quad \text{o aproximadamente } 0.00065536
$$

c. $$ \left(4^{-3}\right)\left[\left(\frac{1}{16}\right)^{-2}\right]^{2} $$

Primero, simplificamos $$ \left(\frac{1}{16}\right)^{-2} $$:
$$
\left(\frac{1}{16}\right)^{-2} = 16^{2} = 256
$$
Ahora, elevamos al cuadrado:
$$
256^{2} = 65536
$$
Multiplicamos:
$$
4^{-3} \cdot 65536 = \frac{1}{64} \cdot 65536 = 1024
$$

d. $$ \left(\frac{5}{3}\right)^{5} \cdot \left(\frac{3}{5}\right)^{5}(3)^{9} $$

Simplificamos:
$$
\left(\frac{5}{3} \cdot \frac{3}{5}\right)^{5} \cdot 3^{9} = 1^{5} \cdot 3^{9} = 3^{9} = 19683
$$

e. $$ \frac{4^{-1}+5^{-1}}{20^{-1}} $$

Calculamos:
$$
4^{-1} = \frac{1}{4}, \quad 5^{-1} = \frac{1}{5}, \quad 20^{-1} = \frac{1}{20}
$$
Por lo tanto:
$$
\frac{\frac{1}{4} + \frac{1}{5}}{\frac{1}{20}} = \left(\frac{5 + 4}{20}\right) \cdot 20 = \frac{9}{20} \cdot 20 = 9
$$

f. $$ \left(\frac{3}{4}\right)^{2} \cdot \left(\frac{9}{16}\right)^{-2} $$

Simplificamos:
$$
\left(\frac{3}{4}\right)^{2} = \frac{9}{16}, \quad \left(\frac{9}{16}\right)^{-2} = \left(\frac{16}{9}\right)^{2} = \frac{256}{81}
$$
Multiplicamos:
$$
\frac{9}{16} \cdot \frac{256}{81} = \frac{16}{9} \quad \text{o } 1\frac{7}{9} \quad \text{o aproximadamente } 1.7
$$

g. $$ 3^{5} \cdot \left[\left(\frac{1}{2}\right)^{2}\right]^{-3} $$

Simplificamos:
$$
\left(\frac{1}{2}\right)^{2} = \frac{1}{4} \quad \Rightarrow \quad \left(\frac{1}{4}\right)^{-3} = 4^{3} = 64
$$
Multiplicamos:
$$
3^{5} \cdot 64 = 243 \cdot 64 = 15552
$$

h. $$ 2^{-5} \cdot \frac{1}{4} \cdot \frac{1}{32} $$

Simplificamos:
$$
\frac{1}{4} = 2^{-2}, \quad \frac{1}{32} = 2^{-5}
$$
Por lo tanto:
$$
2^{-5} \cdot 2^{-2} \cdot 2^{-5} = 2^{-5-2-5} = 2^{-12} = \frac{1}{4096}
$$

Resumiendo los resultados:
- a. $$ \frac{1}{2^{19}} $$
- b. $$ \frac{256}{5^{8}} $$
- c. $$ 1024 $$
- d. $$ 19683 $$
- e. $$ 9 $$
- f. $$ \frac{16}{9} $$
- g. $$ 15552 $$
- h. $$ \frac{1}{4096} $$
a. $$ \frac{1}{2^{19}} $$ b. $$ \frac{256}{5^{8}} $$ c. $$ 1024 $$ d. $$ 19683 $$ e. $$ 9 $$ f. $$ \frac{16}{9} $$ g. $$ 15552 $$ h. $$ \frac{1}{4096} $$

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