Question

${2}^{x - 7} \times {5}^{x - 4} = 1250$
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Answer 1

Answer:

$2^{x-7}\times \:5^{x-4}=1250\quad :\quad x=8$

Step-by-step explanation:

$2^{x-7}\times \:5^{x-4}=1250$

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

$5^{x-4}=5^{x-7}\times \:5^3$

$2^{x-7}\times \:5^{x-7}\times \:5^3=1250$

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

$2^{x-7}\times \:5^{x-7}=\left(2\times \:5\right)^{x-7}=10^{x-7}$

$10^{x-7}\times \:5^3=1250$

$\mathrm{Divide\:both\:sides\:by\:}5^3$

$\frac{10^{x-7}\times \:5^3}{5^3}=\frac{1250}{5^3}$

$\mathrm{Simplify}$

$10^{x-7}=\frac{1250}{5^3}$

$\mathrm{Simplify\:}\frac{1250}{5^3}$

$\mathrm{Factor}\:1250:\quad 5^4\times \:2$

$=\frac{5^4\times \:2}{5^3}$

$\mathrm{Cancel\:}\frac{2\times \:5^4}{5^3}:\quad 2\times \:5$

$=2\times \:5$

$\mathrm{Multiply\:the\:numbers:}\:2\times \:5=10$

$=10$

$10^{x-7}=10$

$\mathrm{If\:}a^{f\left(x\right)}=a^{g\left(x\right)}\mathrm{,\:then\:}f\left(x\right)=g\left(x\right)$

$x-7=1$

$\mathrm{Solve\:}\:x-7=1:\quad x=8$

$x=8$