document.write( "Question 1199910: Find the pair of integers (a,b) for which 2^(a+1) + 2^a = 3^(b+2) - 3^b \n" ); document.write( "

Algebra.Com's Answer #833896 by greenestamps(13330) $\"\"$ $\"About$

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\n" ); document.write( " $\"2%5E%28a%2B1%29%2B2%5Ea=3%5E%28b%2B2%29-3%5Eb\"$

\n" ); document.write( "Simplify using greatest common factors on both sides:

\n" ); document.write( " $\"%282%5Ea%29%282%5E1%2B1%29=%283%5Eb%29%283%5E2-1%29\"$
\n" ); document.write( " $\"%282%5Ea%29%283%29=%283%5Eb%29%288%29\"$
\n" ); document.write( " $\"%282%5Ea%29%283%5E1%29=%283%5Eb%29%282%5E3%29\"$

\n" ); document.write( "Both expressions are prime factorizations, so the exponents on like prime factors must be the same: a=3 and b=1

\n" ); document.write( "ANSWER: (a,b) = (3,1)

\n" ); document.write( "CHECK:
\n" ); document.write( "2^4+2^3 = 16+8 = 24
\n" ); document.write( "3^3-3^1 = 27-3 = 24

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