Question

If \(11^{10-2x} = 1\), find the value of \(x\).

• A. 10
• B. 5
• C. 2
• D. None of these

Hint:

A nonzero base raised to the power 0 always gives 1, so set the exponent \(10-2x\) equal to 0.

Answer 1

The Correct Option is B

Step 1: Understand what makes a power equal to 1.
We are given \(11^{10-2x} = 1\). The base here, 11, is a fixed positive number that is not equal to 1, so the only way a power of 11 can come out to 1 is if the exponent itself is 0. This is a basic rule of exponents: any nonzero number raised to the power 0 equals 1.

Step 2: Set the exponent equal to zero.
Since \(11^{10-2x} = 11^{0}\) is the only way to get 1, we can equate the exponents directly.
This turns the problem into a simple linear equation.
\[ 10 - 2x = 0 \]

Step 3: Solve for x.
Move the constant to the other side.
\[ 2x = 10 \]
Divide both sides by 2.
\[ x = 5 \]

Step 4: Check why the other options fail.
Option (a) 10 gives an exponent of \(10 - 20 = -10\), so \(11^{-10}\) is a very small fraction, not 1.
Option (c) 2 gives an exponent of \(10 - 4 = 6\), so \(11^{6}\) is a huge number, not 1.
Option (d) is wrong because a valid value of x does exist, so it cannot be "none of these".

Final Answer:
The value of x that satisfies the equation is 5. \[ \boxed{x = 5} \]