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Question : 6 of 86

Marks: +1, -0

If

\; \frac{9^{n} \times 3^{5} \times (27)^{3}}{3 \times (81)^{4}} = 27,

then the value of n is

0
2
3
4.

Solution:

Concept:

Express all numbers as powers of 3 to compare exponents.

Explanation:

Write each term in base 3:

9 = 3^2

,

27 = 3^3

,

81 = 3^4

.

Then

9^n = (3^2)^n = 3^{2n}

,

(27)^3 = (3^3)^3 = 3^9

,

(81)^4 = (3^4)^4 = 3^{16}

.

The left side becomes

\frac{3^{2n} \times 3^5 \times 3^9}{3 \times 3^{16}} = \frac{3^{2n+5+9}}{3^{1+16}} = \frac{3^{2n+14}}{3^{17}} = 3^{2n+14-17} = 3^{2n-3}

.

The right side

27 = 3^3

, so

3^{2n-3} = 3^3

.

Equate exponents:

2n - 3 = 3 \implies 2n = 6 \implies n = 3

.

Answer:

n = 3

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