Find the value of n, where n is an integer and 2^{n - 5} × 6^{2n - 4} = 1/(12⁴ × 2)

Solution:

Given, 2^{n - 5} × 6^{2n - 4} = 1/(12⁴ × 2).

n is an integer.

We have to find the value of n.

Considering 2^{n - 5},

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m ÷ aⁿ = a^{m - n}

Here, a = 2

The term 2^{n - 5} can also be written as

2^{n - 5} = 2ⁿ/2⁵

Similarly, 6^{2n - 4} can be written as 6²ⁿ/6⁴

Now, 2^{n - 5} × 6^{2n - 4} = (2ⁿ/2⁵) × (6²ⁿ/6⁴)

= (2ⁿ × 6²ⁿ)/(2⁵ × 6⁴)

So, (2ⁿ × 6²ⁿ)/(2⁵ × 6⁴) = 1/(12⁴ × 2)

12⁴ can be written as (6 × 2)⁴

For any non-zero integers ‘a’ and ‘b’ and whole numbers m and n,

a^m × b^m = (a × b)^m

So, (6 × 2)⁴ = (6)⁴ × (2)⁴

On rearranging,

2ⁿ × 6²ⁿ = (2⁵ × 6⁴)/((6)⁴ × (2)⁴ ×2)

2ⁿ × (6²)ⁿ = (2⁵ × 6⁴)/((6)⁴ × (2)⁵)

2ⁿ × 36ⁿ = 1

(2 × 36)ⁿ = 1

(72)ⁿ = 1

(72)ⁿ = (72)⁰

The bases are equal.

Equating the powers,

n = 0

Therefore, the required value is 0.

✦ Try This: Find the value of n, where n is an integer and 3^{n - 2} × 2^{n - 4} = 1/(3⁴ × 2)

☛ Also Check: NCERT Solutions for Class 7 Maths Chapter 13

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NCERT Exemplar Class 7 Maths Chapter 11 Problem 73

Find the value of n, where n is an integer and 2^{n - 5} × 6^{2n - 4} = 1/(12⁴ × 2)

Summary:

The value of n is 0, where n is an integer and 2^{n - 5} × 6^{2n - 4} = 1/(12⁴ × 2).

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☛ Related Questions:

• Express the following in usual form: 8.01 × 10⁷
• Express the following in usual form: 1.75 × 10⁻³
• Find the value of 2⁵