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If r^2 - 2rs + s^2 = 4, then (r - s)^(6)...

If `r^2 - 2rs + s^2 = 4`, then `(r - s)^(6) `=

A

`-4`

B

`4`

C

`8`

D

64

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start with the equation given: 1. **Given Equation**: \[ r^2 - 2rs + s^2 = 4 \] 2. **Recognizing the Perfect Square**: We can rewrite the left-hand side as a perfect square: \[ (r - s)^2 = 4 \] This is because \( r^2 - 2rs + s^2 \) can be factored as \( (r - s)^2 \). 3. **Taking the Square Root**: To find \( r - s \), we take the square root of both sides: \[ r - s = \pm 2 \] 4. **Finding \( (r - s)^6 \)**: We need to find \( (r - s)^6 \). Using the values we found: \[ (r - s)^6 = (\pm 2)^6 \] Since raising to an even power will eliminate the negative sign, we have: \[ (r - s)^6 = 2^6 \] 5. **Calculating \( 2^6 \)**: Now we calculate \( 2^6 \): \[ 2^6 = 64 \] Thus, the final answer is: \[ (r - s)^6 = 64 \]
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