Expressions & Equations

Scientific Notation

Expressing very large and very small numbers in scientific notation.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Properties of Exponents

Taming Extremely Large and Small Numbers

The Sun is about 93,000,000 miles from Earth. A red blood cell is about 0.000008 meters wide. Before reading on, think about what's awkward about writing either of these numbers out in full — and what a more compact notation might look like.

Definition — Scientific Notation

A way of writing a number as a value between 1 and 10, multiplied by a power of 10: a × 10ⁿ. The exponent n tells how many places the decimal point shifts to recover the original number.

This connects directly to the powers-of-ten place-value shifting from Grade 5: 93,000,000 is 9.3 shifted 7 places to the left, so it's written 9.3 × 10⁷. A tiny number like 0.000008 is 8 shifted 6 places to the right of where it would sit as a value between 1 and 10, giving a negative exponent: 8 × 10⁻⁶.

Worked Example — Converting to Scientific Notation

Write 93,000,000 in scientific notation. Move the decimal point until exactly one nonzero digit remains before it: 9.3. Count the shift: 7 places to the left. So 93,000,000 = 9.3 × 10⁷.

Worked Example — Converting from Scientific Notation

Write 8 × 10⁻⁶ in standard form. A negative exponent of −6 means shifting the decimal 6 places to the right (making the number smaller): 0.000008.

Worked Example — Multiplying Numbers in Scientific Notation

Multiply (3 × 10⁴) × (2 × 10⁶). Multiply the decimal parts: 3 × 2 = 6. Add the exponents, using the product rule from the last lesson: 10⁴⁺⁶ = 10¹⁰. The result is 6 × 10¹⁰.

93{,}000{,}000 = 9.3 \times 10^{7} \qquad 0.000008 = 8 \times 10^{-6}

Tip

A positive exponent in scientific notation always corresponds to a number of 10 or greater; a negative exponent always corresponds to a number between 0 and 1 — a quick check on whether the sign of the exponent matches the size of the original number.

Common Mistakes

Leaving more than one digit before the decimal point in the coefficient, such as writing 93 × 10⁶ instead of 9.3 × 10⁷.
The coefficient in proper scientific notation is always between 1 and 10 — shift the decimal point until only one nonzero digit remains before it, adjusting the exponent to match.
Using the wrong sign on the exponent, such as writing a positive exponent for a number smaller than 1.
Numbers 10 or greater use a positive exponent; numbers between 0 and 1 use a negative exponent — check the original number's size against this rule before finalizing the sign.

Key Takeaways

Scientific notation writes a number as a value between 1 and 10 times a power of 10.
The exponent's sign matches whether the original number is 10 or greater (positive) or between 0 and 1 (negative).
Multiplying numbers in scientific notation multiplies the coefficients and adds the exponents, using the product rule for exponents.

Summary

Scientific notation makes extremely large and small numbers manageable using exponent rules already learned. The next unit shifts from exponents to linear relationships, starting with the single number that describes a line's steepness.

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