How do you multiply #2w ^ { 9} u ^ { 7} \cdot 3w \cdot 2u ^ { 7}#?

First, rewrite the expression as:

#(2 * 3 * 2)(w^9 * w)(u^7 * u^7) =>#

#12(w^9 * w)(u^7 * u^7)#

Now, use these rules of exponents to complete the multiplication of the #w# and #u# terms:

#a = a^color(blue)(1)# and #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#

#12(w^9 * w)(u^7 * u^7) =>#

#12(w^color(red)(9) * w^color(blue)(1))(u^color(red)(7) * u^color(blue)(7)) =>#

#12w^(color(red)(9)+color(blue)(1))u^(color(red)(7)+color(blue)(7)) =>#

#12w^10u^14#

Just imagine yourself multiplying #3w# with #2w^9u^7# and #2u^7# with #2w^9u^7#, then multiplying these together

First let's do the w.

#2w^9*3w^1# (since #3w=3w^1#)

To multiply powers, you apply the following rule

#a^x*a^y=a^(x+y)#

#therefore 2w^9*3w^1=(2*3)w^(9+1)#
#=6w^10#

Same applies to the u.

#u^7*2u^7=2u^(7+7)#
#=2u^14#

Now obviously, #6*2=12#

Combining them, we get #12u^14w^10# (#u# first because it has to be in alphabetical order)

The indices or exponents of a number of variable indicates how many of them are multiplied together.

#x^4 = x*x*x*x" "larr# remember #*# means #xx#

When we multiply in algebra we multiply all the same types of bases together.

#2*w^9u^7*3w*2u^7# can be re-arranged and written as:

#=color(blue)(2*3)*color(purple)(u^7*u^7)*color(forestgreen)(w^9*w^1)#

#=color(blue)(6)*color(purple)(u^14)*color(forestgreen)(w^10)#

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Recall:

Note that:

#color(purple)(u^7*u^7 =u*u*u*u*u*u*u xx u*u*u*u*u*u*u)#

#color(purple)(u^7*u^7 = u^14)" "larr# add the indices of like bases

#color(forestgreen)(w^9*w^1 = w*w*w*w*w*w*w*w*wxxw = w^10#

#color(blue)(2*3=6" "larr" "2 and 3# are just numbers, multiply as usual.