The measurement $78,005,760$ expressed correctly using scientific notation is
$7.8005760 \times 10^{7}$
$7.8005760 \times 10^{-7}$
$7.8 \times 10^{7}$
$7.800576 \times 10^{-7}$
$7.800576 \times 10^{7}$

Step 1 ：First, we need to express the number $78,005,760$ in scientific notation. Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is commonly used by scientists, mathematicians, and engineers.

Step 2 ：In scientific notation, all numbers are written in the form of $a imes 10^n$, where the exponent $n$ is an integer, and the coefficient $a$ is any real number, called the significand or mantissa. The coefficient $a$ must be greater than or equal to 1 and less than 10.

Step 3 ：So, we need to move the decimal point in $78,005,760$ to the right until we get a number between 1 and 10. We find that we need to move the decimal point 7 places to the left to get $7.8005760$.

Step 4 ：Therefore, $78,005,760$ can be expressed in scientific notation as $7.8005760 imes 10^7$.

Step 5 ：Checking the options, we find that $7.8005760 imes 10^7$ is an option. Therefore, the correct answer is \(\boxed{7.8005760 imes 10^7}\).