Express 0.00000345 in standard form

Question

Express $0.00000345$ in standard form.

Solution — Step by Step

Standard form (also called scientific notation) is a way of writing very large or very small numbers in the form:

a \times 10^n

where $1 \leq a < 10$ (a number between 1 and 10) and $n$ is an integer (positive or negative).

For numbers less than 1, $n$ is negative. For numbers greater than 10, $n$ is positive.

We need to write $0.00000345$ as $a \times 10^n$ where $1 \leq a < 10$ .

The non-zero digits start at $3.45$ . We need to move the decimal point to get from $0.00000345$ to $3.45$ .

Count how many places we move the decimal point to the right: $0.00000345 \to 3.45$ requires moving 6 places to the right.

Moving the decimal point 6 places to the right is equivalent to multiplying by $10^6$ . To keep the value unchanged, we must divide by $10^6$ (multiply by $10^{-6}$ ):

0.00000345 = 3.45 \times 10^{-6}

\boxed{3.45 \times 10^{-6}}

$3.45 \times 10^{-6} = 3.45 \div 10^6 = 3.45 \div 1000000 = 0.00000345$ ✓

Why This Works

Every number can be written as a product of a number between 1 and 10, and a power of 10. The power of 10 carries the “scale” information, while the first part carries the significant digits.

The rule for small numbers (less than 1): count how many places you move the decimal point to the right to reach the first significant digit. That count is the magnitude of the negative exponent.

Alternative Method

You can also count the zeros after the decimal point before the first significant digit:

$0.\underbrace{00000}_{5 \text{ zeros}}345$

There are 5 zeros, but the first non-zero digit (3) is in the 6th decimal place. So the exponent is $-6$ .

$3.45 \times 10^{-6}$

Common Mistake

Students sometimes count incorrectly. The exponent is NOT just the number of zeros after the decimal point — it’s the position of the first significant figure from the decimal point. In $0.00000345$ , there are 5 zeros after the decimal, but the 3 is in the 6th position (5 zeros + 1 more place), so the exponent is $-6$ , not $-5$ . Double-check by verifying: $3.45 \times 10^{-6} = 0.00000345$ ✓

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next