Sure, let's go through this step by step.
We need to simplify the expression:
[tex]\[P = \frac{4 x^0 + (2 x)^0 - x^0}{2(x+2)^0 - (\sqrt{7} x)^0}\][/tex]
Let's start by remembering that [tex]\(x^0 = 1\)[/tex] for any non-zero [tex]\(x\)[/tex]. This is a critical property we'll use in our simplification.
### Numerator Simplification:
1. [tex]\( 4 x^0 \)[/tex] becomes [tex]\( 4 \cdot 1 = 4 \)[/tex].
2. [tex]\( (2 x)^0 \)[/tex] becomes [tex]\( (2 \cdot x)^0 = 1 \)[/tex] since any number to the 0 power is 1.
3. [tex]\( x^0 \)[/tex] becomes [tex]\(1\)[/tex].
So, the numerator simplifies as follows:
[tex]\[ 4 x^0 + (2 x)^0 - x^0 = 4 \cdot 1 + 1 - 1 = 4 + 1 - 1 = 4 \][/tex]
### Denominator Simplification:
1. [tex]\( (x + 2)^0 \)[/tex] becomes [tex]\( 1 \)[/tex] because the exponent is 0.
2. [tex]\( \sqrt{7} x \)[/tex] is not 0, but [tex]\( (\sqrt{7} x)^0 \)[/tex] becomes [tex]\( 1 \)[/tex].
So, the denominator simplifies as follows:
[tex]\[ 2(x+2)^0 - (\sqrt{7} x)^0 = 2 \cdot 1 - 1 = 2 - 1 = 1 \][/tex]
### Putting it all together:
Now, let’s combine the simplified results of the numerator and the denominator to find [tex]\(P\)[/tex]:
[tex]\[ P = \frac{\text{Numerator}}{\text{Denominator}} = \frac{4}{1} = 4 \][/tex]
Thus, the value of [tex]\( P \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
Hence, the correct answer is:
C) 4
