Question

simplify (25)^3/2 * (243)^3/5 ÷(625)^1/2 (16)^5/4 * (8)^4/3

Answer 1

Answer:

$25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:16^{\dfrac{5}{4}}\cdot \:8^{\dfrac{4}{3}}=69120$

Step-by-step explanation:

$25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:16^{\dfrac{5}{4}}\cdot \:8^{\dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Factor\:integer\:}16=2^4$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\left(2^4\right)^{\dfrac{5}{4}}\cdot \:8^{\dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(2^4\right)^{\dfrac{5}{4}}=2^{4\cdot \dfrac{5}{4}}$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^{4\cdot \dfrac{5}{4}}\cdot \:8^{\dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Refine}$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^5\cdot \:8^{\dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Factor\:integer\:}8=2^3$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^5\left(2^3\right)^{\dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(2^3\right)^{\dfrac{4}{3}}=2^{3\cdot \dfrac{4}{3}}$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^5\cdot \:2^{3\cdot \dfrac{4}{3}}$

$\rule{300}{1}$

$\mathrm{Refine}$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^5\cdot \:2^4$

$\rule{300}{1}$

$\mathrm{Apply\:exponent\:rule}:\quad \:a^b\cdot \:a^c=a^{b+c}$

$2^5\cdot \:2^4=\:2^{5+4}$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^{5+4}$

$\rule{300}{1}$

$\mathrm{Add\:the\:numbers:}\:5+4=9$

$=25^{\dfrac{3}{2}}\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}\cdot \:2^9$

$\rule{300}{1}$

$25^{\dfrac{3}{2}}$

$\rule{300}{0.5}$

$\mathrm{Factor\:the\:number:\:}\:25=5^2$

$=\left(5^2\right)^{\dfrac{3}{2}}$

$\rule{300}{0.3}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(a^b\right)^c=a^{bc}$

$\left(5^2\right)^{\dfrac{3}{2}}=5^{2\cdot \dfrac{3}{2}}=5^3$

$=5^3$

$\rule{300}{0.3}$

$\mathrm{Refine}$

$=125$

$\rule{300}{1}$

$=2^9\cdot \:125\cdot \dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}$

$\rule{300}{1}$

$\dfrac{243^{\dfrac{3}{5}}}{625^{\dfrac{1}{2}}}$

$\rule{300}{0.3}$

$625^{\dfrac{1}{2}}=25$

$243^{\dfrac{3}{5}}=27$

$\rule{300}{0.3}$

$=\dfrac{27}{25}$

$\rule{300}{1}$

$=2^9\cdot \:125\cdot \dfrac{27}{25}$

$\rule{300}{1}$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \dfrac{b}{c}=\dfrac{a\:\cdot \:b}{c}$

$=\dfrac{27\cdot \:125\cdot \:2^9}{25}$

$\rule{300}{1}$

$\mathrm{Multiply\:the\:numbers:}\:27\cdot \:125=3375$

$=\dfrac{2^9\cdot \:3375}{25}$

$\rule{300}{1}$

$\mathrm{Divide\:the\:numbers:}\:\dfrac{3375}{25}=135$

$=2^9\cdot \:135$

$\rule{300}{1}$

$2^9=512$

$=512\cdot \:135$

$\rule{300}{1}$

$\mathrm{Multiply\:the\:numbers:}\:512\cdot \:135=69120$

$=69120$