Q.34 If 7^3x = 216 , the value of 7^−x (rounded off to three decimal places) is ____________.

Solve 7^3x = 216 and Find 7^−x

To solve the exponential equation 7^3x = 216, we first rewrite 216 into an equivalent exponential form to simplify. The goal is to determine the value of 7^−x rounded to three decimal places.

Step-by-Step Solution

Start with the equation:

73x = 216

Recognize that:

216 = 63

This converts the equation into:

73x = 63

Now raise both sides to the power −1/3 to isolate 7−x:

(73x)−1/3 = (63)−1/3

So:

7−x = 6−1 = 1/6 ≈ 0.1666…

Rounded to three decimal places:

7−x = 0.167

Alternative Method

If solving for x explicitly:

3x = log7(216)

x = log7(216)/3 ≈ 0.129

Then compute 7−x and get the same decimal approximation.

Common Mistakes

• Solving for x then forgetting to compute 7−x.
• Mistaken rounding to 0.160 or 0.170.
• Incorrectly assuming 7−x = 1/7.

Final Answer

7−x = 0.167 (rounded to three decimal places)

Introduction

Exponential equations form a major part of competitive exams like IIT JAM and JEE. The problem “7^3x = 216 value of 7^−x” demonstrates how converting numbers into power form helps solve quickly and accurately.

Detailed Working

Observe that 216 = 63, so:

73x = 63

Raise both sides to −1/3:

7−x = 6−1 = 1/6 = 0.167

Final Value

The required result is 7−x = 0.167.