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§ Algebra

Scientific Notation

§ Algebra

Scientific Notation

CCSS.8.EE3 min read

Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10, written in the form c × 10^n. The number 450,000 becomes 4.5 × 10^5, while 0.0032 becomes 3.2 × 10^-3. This system standardizes number representation across all magnitudes, from atomic scales to astronomical distances.

§ 01

Why it matters

Scientific notation appears throughout science, engineering, and advanced mathematics when dealing with extreme values. Astronomers use it to express distances like 93,000,000 miles (9.3 × 107 miles) from Earth to the Sun. Chemists work with molecular masses such as 6.022 × 1023 particles per mole. Computer scientists measure processing speeds in gigahertz (109 cycles per second). In algebra and calculus, scientific notation simplifies calculations with exponential functions and logarithms. The notation becomes essential in physics courses where students encounter constants like the speed of light (3.0 × 108 meters per second) and Planck's constant (6.626 × 10-34 joule-seconds). Financial modeling also relies on scientific notation for expressing large monetary values in economic equations.

§ 02

How to solve scientific notation

Scientific Notation

  • Write as c × 10n where 1 ≤ c < 10.
  • Count decimal places moved = exponent.
  • Right = negative exponent, left = positive.

Example: 45000 = 4.5 × 10⁴.

§ 03

Worked examples

Beginner§ 01

Write 9000 in scientific notation.

Answer: 9 × 103

  1. Move the decimal point → 9000 = 9 × 103 — Move decimal 3 places left to get 9.
Easy§ 02

Write 790 in scientific notation.

Answer: 7.9 × 102

  1. Find coefficient (1 ≤ c < 10) → 790 = 7.9 × 102 — Coefficient is 7.9, exponent is 2.
Medium§ 03

(3 × 104) × (7 × 102) = _______

Answer: 2.1 × 107

  1. Multiply coefficients, add exponents → 3 × 7 = 21, 104 × 102 = 106 — Coefficients multiply normally, exponents add.
  2. Normalize → 2.1 × 107 — Adjust so coefficient is between 1 and 10.
§ 04

Common mistakes

  • Writing coefficients outside the range 1 to 10, such as expressing 2500 as 25 × 10^2 instead of 2.5 × 10^3
  • Confusing the sign of exponents when converting decimals, writing 0.004 as 4 × 10^3 instead of 4 × 10^-3
  • Adding exponents instead of multiplying coefficients during multiplication, calculating (2 × 10^3) × (3 × 10^4) as 5 × 10^7 instead of 6 × 10^7
§ 05

Frequently asked questions

How do you determine the exponent in scientific notation?
Count the number of decimal places moved to position the decimal point after the first non-zero digit. For numbers greater than 10, the exponent is positive. For numbers less than 1, the exponent is negative. Moving the decimal 4 places left in 85,000 gives an exponent of 4.
What is the difference between 10^3 and 10^-3?
10^3 equals 1,000 (positive exponent means the number is large), while 10^-3 equals 0.001 (negative exponent means the number is small). The negative exponent indicates division by that power of 10, so 10^-3 = 1/10^3 = 1/1000.
How do you multiply numbers in scientific notation?
Multiply the coefficients together and add the exponents. For (4 × 10^6) × (2 × 10^3), multiply 4 × 2 = 8 and add 6 + 3 = 9 to get 8 × 10^9. If the coefficient result exceeds 10, adjust by moving the decimal point and increasing the exponent by 1.
Can the coefficient in scientific notation be negative?
Yes, negative numbers in scientific notation have negative coefficients. The number -6,200 becomes -6.2 × 10^3. The negative sign applies to the coefficient, not the exponent. The exponent still follows the same rules based on the magnitude of the number.
How do you convert scientific notation back to standard form?
Move the decimal point in the coefficient according to the exponent. Positive exponents move the decimal right, negative exponents move it left. For 3.7 × 10^4, move the decimal 4 places right to get 37,000. For 5.2 × 10^-2, move 2 places left to get 0.052.
§ 06

See also

CoefficientExponentRange
§ 06

Where to next?

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