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Scientific Notation — How to Convert and Calculate

Grade: 8-9 | Topic: Arithmetic

What You Will Learn​

After reading this page you will be able to convert any number to and from scientific notation, perform multiplication and division with numbers in scientific notation, and understand why scientists and engineers use this system to handle extremely large and extremely small quantities. This skill appears constantly in science classes and standardized tests.

Theory​

What is scientific notation?​

Scientific notation is a compact way to write very large or very small numbers. Every number in scientific notation has two parts:

a×10na \times 10^{n}

where:

  • aa is the coefficient — a number with exactly one non-zero digit to the left of the decimal point (1≤∣a∣<101 \leq |a| < 10).
  • nn is the exponent — an integer that tells you how many places the decimal point moved.

Examples at a glance:

Standard formScientific notation
93,000,0009.3×1079.3 \times 10^{7}
4,5004.5×1034.5 \times 10^{3}
0.000727.2×10−47.2 \times 10^{-4}
0.00000011×10−71 \times 10^{-7}

Converting a large number to scientific notation​

  1. Place the decimal point after the first non-zero digit.
  2. Count how many places you moved the decimal to the left — that count becomes the positive exponent.

Example: Convert 6,370,000 to scientific notation.

  • Move the decimal: 6.3700006.370000 — moved 6 places left.
  • Result: 6.37×1066.37 \times 10^{6}

Converting a small number to scientific notation​

  1. Place the decimal point after the first non-zero digit.
  2. Count how many places you moved the decimal to the right — that count becomes the negative exponent.

Example: Convert 0.000045 to scientific notation.

  • Move the decimal: 4.54.5 — moved 5 places right.
  • Result: 4.5×10−54.5 \times 10^{-5}

Converting from scientific notation to standard form​

Reverse the process:

  • Positive exponent — move the decimal point to the right.
  • Negative exponent — move the decimal point to the left.

Example: 3.08×104=30,8003.08 \times 10^{4} = 30{,}800

Multiplying in scientific notation​

Multiply the coefficients and add the exponents:

(a×10m)(b×10n)=(a×b)×10m+n(a \times 10^{m})(b \times 10^{n}) = (a \times b) \times 10^{m+n}

If the resulting coefficient is not between 1 and 10, adjust it and change the exponent accordingly.

Dividing in scientific notation​

Divide the coefficients and subtract the exponents:

a×10mb×10n=ab×10m−n\frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n}

Adding and subtracting in scientific notation​

Before adding or subtracting, both numbers must have the same power of 10. Adjust one number so the exponents match, then add or subtract the coefficients.

Worked Examples​

Example 1: Converting a large number​

Problem: Write 149,600,000 in scientific notation.

Step 1: Place the decimal after the first non-zero digit. 1.4961.496

Step 2: Count the places moved left: 8.

Answer: 149,600,000=1.496×108149{,}600{,}000 = \mathbf{1.496 \times 10^{8}}

Example 2: Converting a small number​

Problem: Write 0.0000306 in scientific notation.

Step 1: Move the decimal after the first non-zero digit: 3.063.06.

Step 2: Count the places moved right: 5.

Answer: 0.0000306=3.06×10−50.0000306 = \mathbf{3.06 \times 10^{-5}}

Example 3: Multiplying in scientific notation​

Problem: Calculate (3×104)×(2.5×103)(3 \times 10^{4}) \times (2.5 \times 10^{3}).

Step 1: Multiply the coefficients. 3×2.5=7.53 \times 2.5 = 7.5

Step 2: Add the exponents. 104×103=10710^{4} \times 10^{3} = 10^{7}

Step 3: Combine. Since 1≤7.5<101 \leq 7.5 < 10, no adjustment needed.

Answer: 7.5×107\mathbf{7.5 \times 10^{7}}

Example 4: Dividing in scientific notation​

Problem: Calculate 8.4×1092.1×103\dfrac{8.4 \times 10^{9}}{2.1 \times 10^{3}}.

Step 1: Divide the coefficients. 8.42.1=4\frac{8.4}{2.1} = 4

Step 2: Subtract the exponents. 109−3=10610^{9-3} = 10^{6}

Answer: 4×106\mathbf{4 \times 10^{6}}

Example 5: Adjusting the coefficient after multiplication​

Problem: Calculate (6×105)×(8×103)(6 \times 10^{5}) \times (8 \times 10^{3}).

Step 1: Multiply the coefficients. 6×8=486 \times 8 = 48

Step 2: Add the exponents. 105+3=10810^{5+3} = 10^{8}

Step 3: 4848 is not between 1 and 10, so adjust: 48=4.8×10148 = 4.8 \times 10^{1}. 4.8×101×108=4.8×1094.8 \times 10^{1} \times 10^{8} = 4.8 \times 10^{9}

Answer: 4.8×109\mathbf{4.8 \times 10^{9}}

Common Mistakes​

Mistake 1: Coefficient outside the range 1 to 10

❌ 45,000=45×10345{,}000 = 45 \times 10^{3}

✅ 45,000=4.5×10445{,}000 = 4.5 \times 10^{4}

Why this matters: In proper scientific notation the coefficient must satisfy 1≤∣a∣<101 \leq |a| < 10. A coefficient of 45 violates this rule and will lose marks on tests.

Mistake 2: Wrong exponent sign for small numbers

❌ 0.003=3×1030.003 = 3 \times 10^{3}

✅ 0.003=3×10−30.003 = 3 \times 10^{-3}

Why this matters: A positive exponent makes the number larger, and a negative exponent makes it smaller. Mixing these up changes your answer by millions or more.

Mistake 3: Adding coefficients without matching exponents first

❌ (2.5×104)+(3.1×103)=5.6×104(2.5 \times 10^{4}) + (3.1 \times 10^{3}) = 5.6 \times 10^{4}

✅ Convert 3.1×103=0.31×1043.1 \times 10^{3} = 0.31 \times 10^{4}, then 2.5+0.31=2.81×1042.5 + 0.31 = 2.81 \times 10^{4}

Why this matters: You can only add or subtract coefficients when the powers of 10 are equal, just like you can only add fractions with the same denominator.

Practice Problems​

Try these on your own before checking the answers:

  1. Write 7,200,000 in scientific notation.
  2. Write 0.00089 in scientific notation.
  3. Convert 5.03×1065.03 \times 10^{6} to standard form.
  4. Calculate (4×103)×(3×105)(4 \times 10^{3}) \times (3 \times 10^{5}).
  5. Calculate 9.6×1081.2×105\dfrac{9.6 \times 10^{8}}{1.2 \times 10^{5}}.
Click to see answers
  1. 7,200,000=7.2×1067{,}200{,}000 = 7.2 \times 10^{6} (decimal moves 6 places left)
  2. 0.00089=8.9×10−40.00089 = 8.9 \times 10^{-4} (decimal moves 4 places right)
  3. 5.03×106=5,030,0005.03 \times 10^{6} = 5{,}030{,}000 (move decimal 6 places right)
  4. (4×3)×103+5=12×108=1.2×109(4 \times 3) \times 10^{3+5} = 12 \times 10^{8} = 1.2 \times 10^{9} (adjust coefficient)
  5. 9.61.2×108−5=8×103=8,000\dfrac{9.6}{1.2} \times 10^{8-5} = 8 \times 10^{3} = 8{,}000

Summary​

  • Scientific notation writes a number as a×10na \times 10^{n} where 1≤∣a∣<101 \leq |a| < 10.
  • Large numbers get a positive exponent; small numbers (less than 1) get a negative exponent.
  • To multiply, multiply coefficients and add exponents; to divide, divide coefficients and subtract exponents.
  • To add or subtract, first make the exponents the same, then work with the coefficients.
  • Always adjust the coefficient back to the 1-to-10 range after any calculation.

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