Given the problem [tex]\( x = 10^a \)[/tex], [tex]\( y = 10^b \)[/tex], and [tex]\( x^b y^a = 100 \)[/tex], we want to find the value of [tex]\( ab \)[/tex].
First, let's substitute [tex]\( x \)[/tex] and [tex]\( y \)[/tex] into the equation [tex]\( x^b y^a = 100 \)[/tex]:
[tex]\[ (10^a)^b (10^b)^a = 100 \][/tex]
We can simplify the left-hand side of the equation step by step:
[tex]\[ (10^a)^b = 10^{ab} \][/tex]
[tex]\[ (10^b)^a = 10^{ab} \][/tex]
Therefore, substituting these back, we have:
[tex]\[ 10^{ab} \cdot 10^{ab} = 100 \][/tex]
Combining the exponents on the left-hand side:
[tex]\[ 10^{2ab} = 100 \][/tex]
We know that [tex]\( 100 \)[/tex] can be written as [tex]\( 10^2 \)[/tex]:
[tex]\[ 10^{2ab} = 10^2 \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 2ab = 2 \][/tex]
To solve for [tex]\( ab \)[/tex], we divide both sides by 2:
[tex]\[ ab = 1 \][/tex]
Therefore, the value of [tex]\( ab \)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
