Scientific Notation, Units, and Formulas

Science deals with numbers that are enormous (the distance to the Sun) and tiny (the size of a cell). Writing all those zeros is clumsy and error-prone, so scientists use scientific notation — a compact way to write very large and very small numbers using powers of ten. This lesson also covers unit awareness and using a formula from a reference sheet, two skills that ride along with calculation questions.

What Scientific Notation Looks Like

A number in scientific notation has the form \[ a \times 10^{n}, \] where \(a\) is at least \(1\) and less than \(10\), and \(n\) is an integer. The exponent \(n\) tells you how many places to move the decimal point. For a large number like \(6{,}300{,}000\), you write \(6.3 \times 10^{6}\), because the decimal moves \(6\) places. For a small number like \(0.0000042\), you write \(4.2 \times 10^{-6}\); the negative exponent means the number is less than \(1\).

A quick rule: a positive exponent means a big number, and a negative exponent means a small one. The exponent is just a count of decimal places.

Converting Back and Forth

To turn \(5.1 \times 10^{4}\) into a plain number, move the decimal \(4\) places to the right: \(51{,}000\). To turn \(9.0 \times 10^{-3}\) into a plain number, move the decimal \(3\) places to the left: \(0.009\). Going the other way, count how many places the decimal must move to leave one nonzero digit in front, and that count becomes the exponent.

Units and Formulas

Two habits protect your calculation answers. First, always track units. If a question gives a distance in kilometers and a time in hours, the speed comes out in kilometers per hour — the units tell you the answer is reasonable. Second, use the reference sheet. The test provides common formulas, so you do not need to memorize them all; you need to know which one to use. For example, speed is \[ \text{speed} = \dfrac{\text{distance}}{\text{time}}. \] Plug in the numbers, keep the units attached, and let the calculator handle the arithmetic.

Watch: A Short Video Lesson

mathantics walks through this skill clearly in a few minutes. It is a helpful companion to the reading above:

A Routine for These Questions

1. Write large or small numbers as \(a \times 10^{n}\), with \(1 \le a < 10\).
2. Positive exponent means big; negative exponent means small.
3. To expand, move the decimal by the exponent (right for positive, left for negative).
4. Track units and pull the right formula from the reference sheet.

Practice

1. Write \(72{,}000\) in scientific notation.
2. Write \(0.00058\) in scientific notation.
3. Expand \(3.4 \times 10^{5}\).
4. Expand \(6.0 \times 10^{-4}\).
5. A car travels \(180\) km in \(3\) hours. What is its speed?
6. Does a negative exponent mean a large or a small number?

Answers

1. \(7.2 \times 10^{4}\).
2. \(5.8 \times 10^{-4}\).
3. \(340{,}000\).
4. \(0.0006\).
5. \(\dfrac{180}{3} = 60\) km/hr.
6. A small number (less than 1).

Where This Fits in Your Science Prep

Scientific notation and units support every calculation topic, from ratios and rates to percent change. You will use these skills again in physics topics like motion and speed. See all topics on the Science Topics Hub.