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Question : 52 of 86

Marks: +1, -0

On simplification

\, \frac{2^{x+3} \times 3^{2x-y} \times 5^{x+y+3} \times 6^{y+1}}{6^{x+1} \times 10^{y+3} \times 15^{x}}

reduces to

$-1$
0
1
10 .

Solution:

Concept:

Express all terms as products of prime factors (2, 3, 5) and cancel common factors.

Explanation:

Rewrite each base:

6 = 2 \times 3

,

10 = 2 \times 5

,

15 = 3 \times 5

.

Numerator:

2^{x+3} \times 3^{2x-y} \times 5^{x+y+3} \times (2 \times 3)^{y+1} = 2^{x+3+y+1} \times 3^{2x-y+y+1} \times 5^{x+y+3} = 2^{x+y+4} \times 3^{2x+1} \times 5^{x+y+3}

.

Denominator:

6^{x+1} \times 10^{y+3} \times 15^{x} = (2 \times 3)^{x+1} \times (2 \times 5)^{y+3} \times (3 \times 5)^{x} = 2^{x+1+y+3} \times 3^{x+1+x} \times 5^{y+3+x} = 2^{x+y+4} \times 3^{2x+1} \times 5^{x+y+3}

.

The numerator and denominator are identical, so the fraction equals

1

.

Answer:

1

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