Question

Find the value of the following expression: $\frac {\bigg (2^{19} \times 27{^3} \bigg) + \bigg (15 \times 4^9 \times 9^4 \bigg )} {\bigg( 6^9 \times 2^{10} \bigg) +2^{10}}$

Answer 1

Solution :

[(2^19×27³)+(15×4^9×9⁴ )]/[(6^9×2^10)+2^10]

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i ) 27³ = ( 3³ )³ = 3^9

ii ) 15 = 3 × 5

iii ) 4^9 = ( 2² )^9 = 2^18

iv ) 9⁴ = ( 3² )⁴ = 3^8

v ) 6^9 = ( 2 × 3 )^9 = 2^9 × 3^9
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Now ,

Take numerator

[ (2^19 × 27³ ) + ( 15 × 4^9 × 9⁴ ) ]

= [ ( 2^19 × 3^9 ) + ( 3×5×2^18×3^8 ) ]

= [ 2^19 × 3^9 ] + [ 3^9 × 2^18 × 5 ]

= ( 2^18 × 3^9 ) ( 2 + 5 )

= ( 2^18 × 3^9 × 5 ) -----( 1 )
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Denominator = [ 6^9 × 2^10 + 2^10 ]

= [ ( 2^9 × 3^9 × 2^10 ) + 2^10 ]

= 2^10 × [ 2^9 × 3^9 + 1 ] -----( 2 )
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The value of the expression

= ( 1 )/( 2 )

= ( 2^18 × 3^9 × 5 )/[ 2^10( 2^9 × 3^9 + 1 )

= ( 2^8 × 3^9 × 5 )/( 2^9 × 3^9 + 1 )


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