A massive product!

Let $x$ denote the first and $y$ denote the second factor in the product $P$ .

Then $3.6\cdot { 10 }^{ 18 }<x<3.7\cdot { 10 }^{ 18 }$ , and $3.4\cdot { 10 }^{ 14 }<y<3.5\cdot { 10 }^{ 14 }$ .

Multiplying these inequalities gives $(3.6)(3.4)\cdot { 10 }^{ 32 }<xy<(3.7)(3.5)\cdot { 10 }^{ 32 }$ .

This is equal to $1.224\cdot {10}^{33}<xy<1.295\cdot {10}^{33}$ .

Since the lower bound and upper bound of $xy$ are both 34 digit numbers, $P$ is also a $\boxed { 34 }$ digit number.