Question 189551
I'll do the first three to get you started



# 1



{{{(a-5)^2/(25-a^2)}}} Start with the given expression.



{{{((a-5)(a-5))/(25-a^2)}}} Expand



{{{((a-5)(a-5))/(-(a+5)(a-5))}}} Factor the denominator



{{{-((a-5)(a-5))/((a+5)(a-5))}}} Reduce



{{{-((a-5)highlight((a-5)))/((a+5)highlight((a-5)))}}} Highlight the common terms. 



{{{-((a-5)cross((a-5)))/((a+5)cross((a-5)))}}} Cancel out the common terms. 



{{{-(a-5)/(a+5)}}} Simplify



So {{{(a-5)^2/(25-a^2)=-(a-5)/(a+5)}}} where {{{a<>-5}}} or {{{a<>5}}}




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# 2



{{{(25c+15d)/(50c^2+30d^2)}}} Start with the given expression.



{{{(5(5c+3d))/(50c^2+30d^2)}}} Factor {{{25c+15d}}} to get {{{5(5c+3d)}}}.



{{{(5(5c+3d))/(5(10c^2+6d^2))}}} Factor {{{50c^2+30d^2}}} to get {{{5(10c^2+6d^2)}}}.



{{{(highlight(5)(5c+3d))/(highlight(5)(10c^2+6d^2))}}} Highlight the common terms. 



{{{(cross(5)(5c+3d))/(cross(5)(10c^2+6d^2))}}} Cancel out the common terms. 



{{{(5c+3d)/(10c^2+6d^2)}}} Simplify. 



So {{{(25c+15d)/(50c^2+30d^2)=(5c+3d)/(10c^2+6d^2)}}} 




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# 3



{{{(2w^2+w-6)/(2w+4)}}} Start with the given expression.



{{{((w+2)(2w-3))/(2w+4)}}} Factor {{{2w^2+w-6}}} to get {{{(w+2)(2w-3)}}}.



{{{((w+2)(2w-3))/(2(w+2))}}} Factor {{{2w+4}}} to get {{{2(w+2)}}}.



{{{(highlight((w+2))(2w-3))/(2*highlight((w+2)))}}} Highlight the common terms. 



{{{(cross((w+2))(2w-3))/(2*cross((w+2)))}}} Cancel out the common terms. 



{{{(2w-3)/2}}} Simplify. 



So {{{(2w^2+w-6)/(2w+4)}}} simplifies to {{{(2w-3)/2}}}.



In other words, {{{(2w^2+w-6)/(2w+4)=(2w-3)/2}}} where {{{w<>-2}}}