--- title: "Powers of Ten Notation" description: "Interpret and compare positive numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer; convert between ordinary notation and standard form" canonical: https://lightmysky.com/learn/mathematics/powers-of-ten-notation-mt_bO-njVOige source: https://lightmysky.com/learn/mathematics/powers-of-ten-notation-mt_bO-njVOige.md retrieved: 2026-10-19 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Powers of Ten Notation Interpret and compare positive numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer; convert between ordinary notation and standard form Subject: Mathematics · Area: Number Representation & Place Value · Ages 12 to 14 Page: https://lightmysky.com/learn/mathematics/powers-of-ten-notation-mt_bO-njVOige ## Ready when they can - Convert large and small numbers into standard form A × 10ⁿ - Convert numbers from standard form back to ordinary notation - Compare and order numbers given in standard form ## Lesson: Standard form: A × 10ⁿ You have read large numbers in full and used squares and cubes. Standard form writes a positive number as A × 10ⁿ, with 1 ≤ A < 10. This means A is at least 1 and less than 10. The exponent n is an integer, a number with no fractional part. It may be positive, zero or negative. Zero cannot be written in this form. We also write powers using ^: 10^3 means 10³. *(drawing: For a positive exponent, count the factors of 10. For example, 10³ = 10 × 10 × 10. Each next value is ten times as large.)* **Example.** To write 7,600,000 in standard form, choose 7.6 for A. Moving its decimal point six places right gives 7,600,000, so 7,600,000 = 7.6 × 10⁶. To convert back, multiply 7.6 by 10 six times. In general, multiplying by 10ⁿ for a positive integer n moves the decimal point n places right. Add zeros where needed. A negative power of ten means division: 10⁻⁴ = 1 ÷ 10⁴. Multiplying by 10⁻⁴ means dividing by 10 four times, moving the decimal point four places left. Thus 4.5 × 10⁻⁴ = 0.00045. To write 0.00045 in standard form, move its point four places right to make 4.5, then use exponent -4. With 1 ≤ A < 10, a negative exponent gives a positive value below 1. To compare positive numbers already in standard form, compare the exponents first. The larger exponent gives the larger value because both A values are at least 1 and less than 10. If the exponents match, compare A. To order a mixture of ordinary numbers and standard-form numbers, convert them all to ordinary notation, then compare their place values. Lowering a power of ten by 1 divides its value by 10: 10² = 100, 10¹ = 10, and 10⁰ = 1. A zero exponent does not give zero. Any nonzero number raised to the power 0 equals 1. So A × 10⁰ = A × 1 = A. An expression can have the right value without being in standard form. If A is 10 or more, divide A by 10 and add 1 to the exponent. If 0 < A < 1, multiply A by 10 and subtract 1 from the exponent. Repeat until 1 ≤ A < 10. For example, 27 × 10⁴ = 2.7 × 10⁵. Making one factor ten times smaller and the other ten times larger keeps their product unchanged. To find how much larger one value is, convert both to ordinary numbers and subtract. For example, 6.5 × 10⁴ - 2.5 × 10⁴ = 65,000 - 25,000 = 40,000. To find how many times as large one positive value is as another, convert both to ordinary numbers and divide. For example, 8 × 10⁵ = 800,000 and 4 × 10⁴ = 40,000. Then 800,000 ÷ 40,000 = 20. The larger value is 20 times as large. Ratios of small values use the same method: convert, then divide. For example, 9.6 × 10⁻³ = 0.0096 and 8 × 10⁻⁴ = 0.0008. To calculate 0.0096 ÷ 0.0008, multiply both numbers by 10,000. This keeps the quotient unchanged and gives 96 ÷ 8 = 12. The larger value is 12 times as large. **Recap.** For positive numbers, standard form is A × 10ⁿ with 1 ≤ A < 10 and integer n. Positive powers scale up, negative powers scale down, and 10⁰ = 1. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Integer powers and roots](https://lightmysky.com/learn/mathematics/integer-powers-and-roots-mt_hCVPYlF-7Y) - [Numbers to ten million](https://lightmysky.com/learn/mathematics/numbers-to-ten-million-mt_XNmGwNggdU) ## Opens up - [Standard Form](https://lightmysky.com/learn/mathematics/standard-form-mt_GbZZ2xjBTt) - [Binary Numbers and Place Value](https://lightmysky.com/learn/computing/binary-numbers-and-place-value-mt_t67akIKBmI)