The number of the real roots of the equation $3^{2x^2-7x+7}=9$ is:

$3^{2x^2-7x+7}=9$

$\Rightarrow 3^{2x^2-7x+7}=3^2$

Comparing both the sides, we get:

⇒ 2x² − 7x + 7 = 2

⇒ 2x² − 7x + 5 = 0

⇒ 2x² − 2x − 5x + 5 = 0

⇒ 2x(x − 1) − 5 (x − 1) = 0

⇒ (x − 1)(2x − 5) = 0

⇒ x = 1, x = 5/2

Hence, the number of real roots is 2.