SL 1.1—Using standard form

The number $0.0000785$ is less than 1, so we move the decimal point right until we obtain a number $a$ where $1\le a<10$.

Moving the decimal point right: $0.0000785\to 7.85$. We moved 5 places to the right.

Because we moved right, the exponent is negative: $n=-5$. So $0.0000785=7.85\times 10^{-5}$.

Check: $a=7.85$ satisfies $1\le 7.85<10$ ✓, and $n=-5$ is an integer ✓. The answer is $7.85\times 10^{-5}$.

Standard form requires $a\times 10^n$ where $1\le a<10$ and $n$ is an integer. We need to check each option for these conditions.

This equals $785,000$, which is a large number — not $0.0000785$. The sign of the exponent is wrong for a number less than 1.

$0.785\times 10^{-4}=0.0000785$ numerically, but $a=0.785$ does not satisfy $1\le a<10$, so this is not in standard form.

$78.5\times 10^{-6}=0.0000785$ numerically, but $a=78.5\ge 10$, so this is not in standard form.

$7.85\times 10^{-5}$ equals $0.0000785$, and $a=7.85$ satisfies $1\le a<10$. This is the only valid standard form expression.

The exponent is $-3$, which is negative. A negative exponent means we move the decimal point left, making the number smaller.

Starting from $2.04$, move the decimal point 3 places to the left: $2.04\to 0.00204$.

$2.04\times 10^{-3}=0.00204$. We can verify: $0.00204\times 1000=2.04$ ✓.

Since the exponent $-3$ is negative, the number must be less than 1. Any option greater than or equal to 1 can be immediately eliminated.

$2040$ is much greater than 1. A negative exponent cannot produce a large number from a value of $a$ between 1 and 10.

$0.0204=2.04\times 10^{-2}$, not $10^{-3}$. The decimal has only moved 2 places, not 3.

$0.000204=2.04\times 10^{-4}$. The decimal has moved 4 places, not 3.

$0.00204$ corresponds to moving the decimal 3 places left from $2.04$, confirming $2.04\times 10^{-3}=0.00204$.