How do you simplify the expression #(8e^-4f^-2)/(18ef^-5)# using the properties?

Question

Julia Adams · Accepted Answer

See a solution process below:

first, simplify the constants as: #(8e^-4f^-2)/(18ef^-5) = ((2 xx 4)e^-4f^-2)/((2 xx 9)ef^-5) =# #((color(red)(cancel(color(black)(2))) xx 4)e^-4f^-2)/((color(red)(cancel(color(black)(2))) xx 9)ef^-5) = (4e^-4f^-2)/(9ef^-5)# Next, use these rules of exponents to simplify the #e# terms: #a = a^color(red)(1)# and #x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))# #(4e^-4f^-2)/(9ef^-5) = (4e^color(red)(-4)f^-2)/(9e^color(red)(1)f^-5) = # #(4f^-2)/(9e^(color(red)(1)-color(red)(-4))f^-5) = (4f^-2)/(9e^(color(red)(1)+color(red)(4))f^-5) =# #(4f^-2)/(9e^5f^-5)# Now, use this rule of exponents to simplify the #f# term: #x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))# #(4f^color(red)(-2))/(9e^5f^color(blue)(-5)) = (4f^(color(red)(-2)-color(blue)(-5)))/(9e^5) =# #(4f^(color(red)(-2)+color(blue)(5)))/(9e^5) = (4f^3)/(9e^5)#

Eva Adamson · Answer

undefined To simplify the expression $ \frac{8e^{-4}f^{-2}}{18ef^{-5}} $, you can follow these steps:

1. Combine the constants: $ \frac{8}{18} = \frac{4}{9} $.
2. Combine the $ e $ terms: $ e^{-4} $ in the numerator and $ e $ in the denominator result in $ e^{-4 - 1} = e^{-5} $.
3. Combine the $ f $ terms: $ f^{-2} $ in the numerator and $ f^{-5} $ in the denominator result in $ f^{-2 - (-5)} = f^{3} $.

So, the simplified expression is $ \frac{4e^{-5}f^{3}}{9} $.