Math, asked by BhavyaNanda2003, 1 year ago

simplify a^4-(a-b)^4

Answers

Answered by MaheswariS
2

\underline{\textbf{Given:}}

\mathsf{a^4-(a-b)^4}

\underline{\textbf{To find:}}

\mathsf{Factors\;of\;a^4-(a-b)^4}

\underline{\textbf{Solution:}}

\mathsf{Consider,}

\mathsf{a^4-(a-b)^4}

\textsf{This can be written as}

\mathsf{=(a^2)^2-((a-b)^2)^2}

\textsf{Using the identity,}\;\boxed{\bf\,a^2-b^2=(a-b)(a+b)}

\mathsf{=(a^2-(a-b)^2)\,(a^2+(a-b)^2)}

\textsf{Using the identities,}

\;\boxed{\bf\,a^2-b^2=(a-b)(a+b)\;\;\&\;\;(a-b)^2=a^2+b^2-2ab}

\mathsf{=(a-(a-b))\,(a+(a-b))\,(a^2+a^2+b^2-2ab)}

\mathsf{=(a-a+b)\,(a+a-b)\,(a^2+a^2+b^2-2ab)}

\mathsf{=b\,(2a-b)\,(2a^2+b^2-2ab)}

\implies\boxed{\boxed{\bf\,a^4-(a-b)^4=b\,(2a-b)\,(2a^2+b^2-2ab)}}

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