Question

Find the value of the following expression: $\frac {\bigg (8 \times (4^2)^4 \times 3^3 \times 2^{7^2} \bigg) + \bigg (9 \times 6^3 \times 4^7 \times (3^2)^{3}\bigg )} {\bigg( 24 \times (6^2)^{4} \times (2^4)^{2}\bigg) + \bigg(144 \times (2^3)^{4} \times (9^2)^{2} \times 4^2 \bigg)}$

Answer 1

Solution :

i ) 8 × ( 4² )² × 3³ × 2^7^2

= 2³ × 4⁴ × 3³ × 2^49

= 2³ × ( 2² )⁴ × 3³ × 2^49

= 2³ × 2^8 × 3³ × 2^49

= 2^3+8+49 × 3³

= 2^62 × 3³ -----( 1 )

ii ) 9 × 6³ × 4^7 × ( 3² )³

= 3² × ( 2 × 3 )³ × ( 2² )^7 × 3^6

= 3² × 2³ × 3³ × 2^14 × 3^6

= 2^3+14 × 3^2+3+6

= 2^17 × 3^11 ----( 2 )

iii ) 24 × ( 6² )⁴ × ( 2⁴ )²

= ( 2³ × 3¹ ) × 6^8 × 2^8

= 2³ × 3¹ × ( 2 × 3 )^8 × 2^8

= 2³ × 3¹ × 2^8 × 3^8 × 2^8

= 2^3+8+8 × 3^1+8

= 2^19 × 3^9 ------( 3 )

iv ) 144 × ( 2³ )⁴ × ( 9² )² × 4²

= 2⁴ × 3² × 2^12 × 9⁴ × ( 2² )²

= 2⁴ × 3² × 2^12 × ( 3² )⁴ × 2⁴

= 2⁴ × 3² × 2^12 × 3^8 × 2⁴

= 2^4+12+4 × 3^2+8

= 2^20 × 3^10 -----( 4 )

______________________

Now ,

Value of the expression

= [ ( 1 ) + ( 2 ) ]/[ ( 3 ) + ( 4 ) ]

=[(2^62×3³)+(2^17×3^11 )]/[(2^19×3^9)+(2^20×3^10)]

= [(2^17×3³){2^45+3^8}]/[(2^19×3^9){1+2×3}]

= [ 2^45 + 3^8 ]/[ ( 2²×3^6 )×7 ]

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