Question 190660


{{{81m^4-625}}} Start with the given expression.



{{{(9m^2)^2-625}}} Rewrite {{{81m^4}}} as {{{(9m^2)^2}}}.



{{{(9m^2)^2-(25)^2}}} Rewrite {{{625}}} as {{{(25)^2}}}.



Notice how we have a difference of squares {{{A^2-B^2}}} where in this case {{{A=9m^2}}} and {{{B=25}}}.



So let's use the difference of squares formula {{{A^2-B^2=(A+B)(A-B)}}} to factor the expression:



{{{A^2-B^2=(A+B)(A-B)}}} Start with the difference of squares formula.



{{{(9m^2)^2-(25)^2=(9m^2+25)(9m^2-25)}}} Plug in {{{A=9m^2}}} and {{{B=25}}}.



So this shows us that {{{81m^4-625}}} factors to {{{(9m^2+25)(9m^2-25)}}}.



In other words {{{81m^4-625=(9m^2+25)(9m^2-25)}}}.



Now let's factor {{{9m^2-25}}}



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{{{9m^2-25}}} Start with the given expression.



{{{(3m)^2-25}}} Rewrite {{{9m^2}}} as {{{(3m)^2}}}.



{{{(3m)^2-(5)^2}}} Rewrite {{{25}}} as {{{(5)^2}}}.



Notice how we have a difference of squares {{{A^2-B^2}}} where in this case {{{A=3m}}} and {{{B=5}}}.



So let's use the difference of squares formula {{{A^2-B^2=(A+B)(A-B)}}} to factor the expression:



{{{A^2-B^2=(A+B)(A-B)}}} Start with the difference of squares formula.



{{{(3m)^2-(5)^2=(3m+5)(3m-5)}}} Plug in {{{A=3m}}} and {{{B=5}}}.



So this shows us that {{{9m^2-25}}} factors to {{{(3m+5)(3m-5)}}}.



In other words {{{9m^2-25=(3m+5)(3m-5)}}}.


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So {{{81m^4-625}}} completely factors to {{{(9m^2+25)(3m+5)(3m-5)}}}



In other words, {{{81m^4-625=(9m^2+25)(3m+5)(3m-5)}}}