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

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

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  1. [tex]\large\underline{\sf{Solution-}}[/tex]

    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!!

    [tex]\rm :\longmapsto\:Prime \: factorization \: of \: 686[/tex]

    [tex]\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}[/tex]

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

    [tex]\rm :\longmapsto\:Prime \: factorization \: of \: 112[/tex]

    [tex]\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}[/tex]

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

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

    [tex]\rm :\longmapsto\: \sf \: = \: {2}^{5} \times {7}^{4} [/tex]

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