### Instruction: When $(10^5 \cdot 10^{77} \div 10^{15} )\div \left(10^{15}\right)^4$ is written as a single integer with no exponents, it is a 1 followed by one or more zeroes. How many zeroes are there?### Response: I need to simplify the expression using the rules of exponents and division.
First, I'll use the fact that $10^a \cdot 10^b = 10^{a+b}$ to combine the powers of 10 in the numerator.
I get $(10^5 \cdot 10^{77} \div 10^{15} )\div \left(10^{15}\right)^4 = (10^{82} \div 10^{15}) \div \left(10^{15}\right)^4$.
Next, I'll use the fact that $10^a \div 10^b = 10^{a-b}$ to simplify the division in the numerator.
I get $(10^{82} \div 10^{15}) \div \left(10^{15}\right)^4 = 10^{67} \div \left(10^{15}\right)^4$.
Now, I'll use the fact that $(10^a)^b = 10^{ab}$ to rewrite the power of 10 in the denominator.
I get $10^{67} \div \left(10^{15}\right)^4 = 10^{67} \div 10^{60}$.
Finally, I'll use the same rule of division to simplify the final division.
I get $10^{67} \div 10^{60} = 10^{67 - 60} = 10^7$.
This means that the expression is equal to 10,000,000, which is a 1 followed by 7 zeroes.
Therefore, the answer is 7.