Math, asked by yadhunandpr45, 3 months ago

Express 686×112 as a product of prime factors only in exponential form​

Answers

Answered by mathdude500
1

\large\underline{\sf{Solution-}}

Basic Concept :-

  • Prime Factorization" is finding which prime numbers multiply together to make the original number.

How to find prime factorization :-

  • Step 1: Start by dividing the number by the first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Then divide by 3, 5, 7, etc. until the only numbers left are prime numbers.

  • Step 2: Write the number as a product of prime numbers.

Let's solve the problem now!!

\rm :\longmapsto\:Prime  \: factorization \:  of  \: 686

\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:686\:\:\:}}}\\ {\underline{\sf{7}}}& \underline{\sf{\:\:343\:\:\:}} \\\underline{\sf{7}}&\underline{\sf{\:\:49\: \:\:}}\\{\underline{\sf{7}}}&{\underline{\sf{\:\:7\:\:\:}}} \\{\underline{\sf{}}}&{{\sf{\:\:1\:\:\:}}}  \end{array}\end{gathered}\end{gathered}\end{gathered}

\rm :\implies\:Prime \:  factorization \:  of  \: 686 = 2 \times  {7}^{3}

\rm :\longmapsto\:Prime  \: factorization  \: of \:  112

\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:112\:\:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:56\:\:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:28\: \:\:}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:14\:\:\:}}} \\{\underline{\sf{7}}}&{\underline{\sf{\:\:7\:\:\:}}}  \\ {\underline{\sf{}}}&{{\sf{\:\:1\:\:\:}}}  \end{array}\end{gathered}\end{gathered}\end{gathered}

\rm :\implies\:Prime \:  factorization \:  of \:  112 =  {2}^{4}  \times 7

 \sf \:  \therefore \: Prime \:  factorization \:  of  \: 686 \:  \times  \: 112 = 2 \times  {7}^{3}  \times  {2}^{4}  \times 7

\rm :\longmapsto\: \sf \:  =  \:  {2}^{5}  \times  {7}^{4}

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