Question #73eb1

To make things easier, you need to know that:

#a^(-n)= 1/a^n#

Using this and starting with the numerator:

#1/4^3 xx 3^4 xx4^2#

#4^3= 4xx4xx4=64#

#3^4= 3xx3xx3xx3= 81#

#4^2= 4xx4=16#

So our numerator is now:

#1/64xx81xx16=1296/64=20.25#

Our denominator is:

#3^5xx1/4^2#

#3^5= 3xx3xx3xx3xx3=243#

#4^2=16color(white(*)# (from above)

So we have:

#243/16=15.1875# for the denominator.

Put together:

#20.25/15.1875= 1 1/3#

Some things to keep in mind are:

1. an exponent means the number multiplied by itself, for example #4^3# = #4*4*4#,
2. a negative exponent is the inverse of the number multiplied by itself: #4^-3 = 1/4*1/4*1/4#
3. any number divided by itself is one: #3/3=1#.

Using this we can first expand the numerator (here I am assuming the question is asking #(4^-3 * 3^4 * 4^2) /( 3^5 * 4^-2)# rather than #4^-3 * 3^4 * 4^2/3^5 * 4^-2#)
#4^-3 * 3^4 * 4^2# expands to:
#1/4*1/4*1/4*3*3*3*3*4*4#
or
#(3*3*3*3*4*4)/(4*4*4)#

Knowing that any number divided by itself = 1, this becomes
#(3*3*3*3*cancel(4)*cancel(4))/(4*cancel(4)*cancel(4))#
or
#3^4*4^-1#

We can see that with the 4's, multiplying #4^2 * 4^-3# gave us #4^-1#, showing that we can find the result by adding the exponents! Likewise, dividing by an exponent is the same as subtracting, because #1/3^5# is the same as #3^-5#.

so:
#(4^-3 * 3^4 * 4^2) /( 3^5 * 4^-2)#

#=(3^4*4^-1)/( 3^5 * 4^-2)#

#=3^4*4^-1* 3^-5 * 4^2#

#=3^4* 3^-5*4^-1*4^2#

#=3^-1*4^1#

#=4^1/3^1#

#=4/3#