Multiplying in Scientific Notation

Learn to multiply numbers in scientific notation: multiply the coefficients, add the exponents and normalize the result.

Advanced25 min18 problems
Practice Problems

Definition

To multiply two numbers in scientific notation, you work with the two parts separately: the coefficients and the powers of ten.
(a×10m)×(b×10n)=(a×b)×10m+n(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}
The rule has three moves:
  1. 1.Multiply the coefficients: a×ba \times b
  2. 2.Add the exponents (product rule 10m×10n=10m+n10^m \times 10^n = 10^{m+n})
  3. 3.Normalize: if a×ba \times b is 1010 or more, move the decimal point one place to the left and add 11 to the exponent
Example: (4×103)(3×105)=12×108=1.2×109(4 \times 10^3)(3 \times 10^5) = 12 \times 10^8 = 1.2 \times 10^9
Because both coefficients are between 11 and 1010, their product is always between 11 and 100100. So you never have to move the decimal point more than one place.

Try it now

Which is the product (2×103)(3×104)(2 \times 10^3)(3 \times 10^4)?

Worked Examples

Multiply (2×103)×(4×105)(2 \times 10^3) \times (4 \times 10^5).
1

Group the coefficients and the powers of ten

(2×4)×(103×105)(2 \times 4) \times (10^3 \times 10^5) → Two separate products
2

Multiply the coefficients

2×4=82 \times 4 = 8
3

Add the exponents

103×105=103+5=10810^3 \times 10^5 = 10^{3+5} = 10^8
4

Check the coefficient

1≤8<101 \le 8 < 10, so nothing to fix → 8×1088 \times 10^8

Common Mistakes

Multiplying the exponents: 103×105=101510^3 \times 10^5 = 10^{15}

Why it's wrong: When you multiply powers with the same base, you count the factors of 1010: three tens and five tens make eight tens, not fifteen.

Correct: Add the exponents: 103×105=103+5=10810^3 \times 10^5 = 10^{3+5} = 10^8. You multiply exponents only for a power of a power, like (103)2=106(10^3)^2 = 10^6.

Stopping at 12×10812 \times 10^8

Why it's wrong: The value is right, but 1212 is not between 11 and 1010, so the answer is not in scientific notation yet.

Correct: Normalize: 12×108=1.2×10912 \times 10^8 = 1.2 \times 10^9.

Normalizing in the wrong direction: 12×108=1.2×10712 \times 10^8 = 1.2 \times 10^7

Why it's wrong: Making the coefficient 1010 times smaller must be balanced by making the power of ten 1010 times bigger, otherwise the value changes.

Correct: Coefficient smaller, exponent bigger: 12×108=1.2×10912 \times 10^8 = 1.2 \times 10^9.

Losing the minus sign: 10−4×107=10−1110^{-4} \times 10^7 = 10^{-11} or 101110^{11}

Why it's wrong: The exponents are signed numbers, and adding them follows the rules for integers.

Correct: −4+7=3-4 + 7 = 3, so 10−4×107=10310^{-4} \times 10^7 = 10^3. Write the addition out with brackets when signs mix: (−4)+7(-4) + 7.

Interactive Visual

Multiply step by step: 12 × 10^8 still has to be normalized

Multiply
(4×103)×(3×105)(4 \times 10^{3}) \times (3 \times 10^{5})

  1. 1Group the coefficients and the powers of 10.

    (4×3)×(103×105)(4 \times 3) \times (10^{3} \times 10^{5})

Multiply the coefficients. What do you get?

Exponents as repeated factors

Product rule

Same base, multiply: add the exponents.

103⋅10510^{3} \cdot 10^{5}

Write each power as repeated factors

1010·1010·1010
10310^{3}3 factors
·
1010·1010·1010·1010·1010
10510^{5}5 factors

3 + 5 = 8 factors in all

Result

103⋅105\displaystyle 10^{3} \cdot 10^{5}==103+5\displaystyle 10^{3 + 5}==108\displaystyle 10^{8}

Value: 108=10000000010^{8} = 100000000

Your turn: change the numbers with the sliders, or try new numbers

Multiply
(2.5×10−4)×(6×107)(2.5 \times 10^{-4}) \times (6 \times 10^{7})

  1. 1Group the coefficients and the powers of 10.

    (2.5×6)×(10−4×107)(2.5 \times 6) \times (10^{-4} \times 10^{7})

Multiply the coefficients. What do you get?

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which is the product (2×103)(3×104)(2 \times 10^3)(3 \times 10^4)?

Why It Matters

Science is full of multiplications with huge or tiny numbers, and scientific notation turns them into quick mental math:
  • Astronomy: light travels 3×1053 \times 10^5 km every second. Multiply by the seconds in a year and you get the length of a light-year
  • Biology: a sample has 4×1064 \times 10^6 bacteria in each milliliter. How many are in a whole flask?
  • Computing: 4×1044 \times 10^4 photos of 3×1063 \times 10^6 bytes each fill 1.2×10111.2 \times 10^{11} bytes of memory
  • Chemistry: number of molecules times the mass of one molecule gives the mass of a sample
Writing these numbers out in full means counting dozens of zeros, and one missing zero makes the answer ten times too small. With scientific notation you multiply two small numbers and add two exponents.

Real World Applications

Astronomy: How Long Is a Light-Year?

A light-year is the distance light travels in one year. Astronomers get it by multiplying the speed of light by the time, both written in scientific notation.

Example:

In 500=5×102500 = 5 \times 10^2 seconds light travels (3×105)(5×102)=15×107=1.5×108(3 \times 10^5)(5 \times 10^2) = 15 \times 10^7 = 1.5 \times 10^8 km, about the distance from the Sun to the Earth.

1Try It Yourself

Light travels 3×1053 \times 10^5 km per second. One year has about 3.15×1073.15 \times 10^7 seconds.

How many kilometers does light travel in one year? (distance = speed × time)

Step 1: Write the mathematical expression

Multiply the speed by the time:

Biology: Counting Bacteria

Microbiologists count the bacteria in a small sample and multiply by the volume to estimate the total. The numbers grow so fast that scientific notation is the only practical way to write them.

Example:

A drop of 5×10−25 \times 10^{-2} mL from a culture with 2×1062 \times 10^6 bacteria per mL holds (2×106)(5×10−2)=10×104=1×105(2 \times 10^6)(5 \times 10^{-2}) = 10 \times 10^4 = 1 \times 10^5 bacteria.

2Try It Yourself

A culture contains 4×1064 \times 10^6 bacteria in each milliliter. The flask holds 5×1025 \times 10^2 mL.

How many bacteria are in the whole flask?

Step 1: Write the mathematical expression

Multiply the bacteria per mL by the number of mL:

Computing: Filling a Memory Card

File sizes and storage are measured in bytes, and the numbers quickly reach billions. Engineers multiply file size by the number of files to plan how much storage a device needs.

Example:

2×1032 \times 10^3 songs of 4×1064 \times 10^6 bytes each take 8×1098 \times 10^9 bytes, that is 8 gigabytes.

3Try It Yourself

One photo takes about 3×1063 \times 10^6 bytes. A phone stores 4×1044 \times 10^4 photos.

How many bytes do the photos take in total?

Step 1: Write the mathematical expression

Multiply the size of one photo by the number of photos:

Key Takeaways

  • 1To multiply in scientific notation, multiply the coefficients and multiply the powers of ten separately
  • 2Powers of ten are multiplied by adding the exponents: 10m×10n=10m+n10^m \times 10^n = 10^{m+n}
  • 3If the new coefficient is 1010 or more, normalize: move the decimal point one place left and add 11 to the exponent
  • 4With negative exponents, add the signed numbers carefully: 10−4×107=10310^{-4} \times 10^7 = 10^3
  • 5A power works on both parts: (a×10m)k=ak×10m×k(a \times 10^m)^k = a^k \times 10^{m \times k}
  • 6Check your answer: the coefficient must satisfy 1≤a<101 \le a < 10

Frequently Asked Questions

An exponent counts how many tens are multiplied. 103×10510^3 \times 10^5 is three tens times five tens, which is 3+5=83 + 5 = 8 tens in a row: 10810^8.
An exponent counts how many tens are multiplied. 103×10510^3 \times 10^5 is three tens times five tens, which is 3+5=83 + 5 = 8 tens in a row: 10810^8.
Yes, the value is the same. But with numbers like 3×105×3.15×1073 \times 10^5 \times 3.15 \times 10^7 you would multiply 300,000300{,}000 by 31,500,00031{,}500{,}000 and count many zeros. In scientific notation you multiply 3×3.153 \times 3.15 and add 5+75 + 7.
Not when you multiply two numbers in scientific notation: both coefficients are less than 1010, so the product is less than 10×10=10010 \times 10 = 100. You move the decimal point at most one place. With powers or with three or more factors it can be bigger, so always check.
Calculators write scientific notation with an E: 1.2E91.2\text{E}9 means 1.2×1091.2 \times 10^9. The number after the E is the exponent.

Glossary

Scientific notation
A number written as a×10na \times 10^n with 1≤a<101 \le a < 10 and nn an integer
Coefficient
The number aa in front of the power of ten, between 11 and 1010
Exponent
The integer nn in 10n10^n; it tells how many places the decimal point moves
Product rule
am×an=am+na^m \times a^n = a^{m+n}: multiplying powers with the same base adds the exponents
Normalize
Rewrite a result like 12×10812 \times 10^8 so that the coefficient is between 11 and 1010: 1.2×1091.2 \times 10^9

Formula Card

Multiplying in scientific notation

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

Multiply the coefficients, add the exponents.

Normalizing

12×108=1.2×10912 \times 10^8 = 1.2 \times 10^9

Coefficient divided by $10$, exponent plus $1$. The value does not change.

Power of a number in scientific notation

(a×10m)k=ak×10m×k(a \times 10^m)^k = a^k \times 10^{m \times k}

Raise the coefficient to the power, multiply the exponent by $k$, then normalize.

Today's challengeOne problem, three tries

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