A Product That Is Already Normalized

Multiply $(2 \times 10^3) \times (4 \times 10^5)$.

Group the coefficients and the powers of ten: $(2 \times 4) \times (10^3 \times 10^5)$ = Two separate products

Multiply the coefficients: $2 \times 4 = 8$ = $8$

Add the exponents: $10^3 \times 10^5 = 10^{3+5} = 10^8$ = $10^8$

Check the coefficient: $1 \le 8 < 10$, so nothing to fix = $8 \times 10^8$

Answer: $(2 \times 10^3)(4 \times 10^5) = 8 \times 10^8$

A Product That Needs Normalizing

Multiply $(4 \times 10^3) \times (3 \times 10^5)$.

Multiply the coefficients: $4 \times 3 = 12$ = $12$

Add the exponents: $3 + 5 = 8$ = $12 \times 10^8$

Is the coefficient between 1 and 10?: $12 \ge 10$, so this is not scientific notation yet = Normalize

Write $12$ as $1.2 \times 10^1$: $12 \times 10^8 = 1.2 \times 10^1 \times 10^8$ = $1.2 \times 10^9$

Answer: $(4 \times 10^3)(3 \times 10^5) = 1.2 \times 10^9$

Two Negative Exponents

Multiply $(2 \times 10^{-5}) \times (7 \times 10^{-3})$.

Multiply the coefficients: $2 \times 7 = 14$ = $14$

Add the exponents (signs included): $-5 + (-3) = -8$ = $14 \times 10^{-8}$

Normalize the coefficient: $14 = 1.4 \times 10^1$ = Add $1$ to the exponent

Combine: $-8 + 1 = -7$ = $1.4 \times 10^{-7}$

Answer: $(2 \times 10^{-5})(7 \times 10^{-3}) = 1.4 \times 10^{-7}$

A Negative and a Positive Exponent

Multiply $(2.5 \times 10^{-4}) \times (6 \times 10^7)$.

Multiply the coefficients: $2.5 \times 6 = 15$ = $15$

Add the exponents: $-4 + 7 = 3$ = $15 \times 10^3$

Normalize: $15 = 1.5 \times 10^1$, so the exponent becomes $3 + 1 = 4$ = $1.5 \times 10^4$

Check with ordinary numbers: $0.00025 \times 60{,}000{,}000 = 15{,}000$ = $1.5 \times 10^4 = 15{,}000$

Answer: $(2.5 \times 10^{-4})(6 \times 10^7) = 1.5 \times 10^4$

Raising to a Power

Calculate $(5 \times 10^3)^3$.

Raise each factor to the power: $(5 \times 10^3)^3 = 5^3 \times (10^3)^3$ = Power of a product

Cube the coefficient: $5^3 = 5 \times 5 \times 5 = 125$ = $125$

Multiply the exponents (power of a power): $(10^3)^3 = 10^{3 \times 3} = 10^9$ = $125 \times 10^9$

Normalize: $125 = 1.25 \times 10^2$, so the exponent becomes $9 + 2 = 11$ = $1.25 \times 10^{11}$

Answer: $(5 \times 10^3)^3 = 1.25 \times 10^{11}$. With a power the coefficient can reach $100$ or more, so you may have to move the decimal point two places.

Mistake: Multiplying the exponents: $10^3 \times 10^5 = 10^{15}$

Why: When you multiply powers with the same base, you count the factors of $10$: three tens and five tens make eight tens, not fifteen.

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

Mistake: Stopping at $12 \times 10^8$

Why: The value is right, but $12$ is not between $1$ and $10$, so the answer is not in scientific notation yet.

Correct: Normalize: $12 \times 10^8 = 1.2 \times 10^9$.

Mistake: Normalizing in the wrong direction: $12 \times 10^8 = 1.2 \times 10^7$

Why: Making the coefficient $10$ times smaller must be balanced by making the power of ten $10$ times bigger, otherwise the value changes.

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

Mistake: Losing the minus sign: $10^{-4} \times 10^7 = 10^{-11}$ or $10^{11}$

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

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

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.

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

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.

A drop of $5 \times 10^{-2}$ mL from a culture with $2 \times 10^6$ bacteria per mL holds $(2 \times 10^6)(5 \times 10^{-2}) = 10 \times 10^4 = 1 \times 10^5$ bacteria.

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.

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

Q: Why do we add the exponents instead of multiplying them?

A: An exponent counts how many tens are multiplied. $10^3 \times 10^5$ is three tens times five tens, which is $3 + 5 = 8$ tens in a row: $10^8$.

Q: Could I just write the numbers out and multiply?

A: Yes, the value is the same. But with numbers like $3 \times 10^5 \times 3.15 \times 10^7$ you would multiply $300{,}000$ by $31{,}500{,}000$ and count many zeros. In scientific notation you multiply $3 \times 3.15$ and add $5 + 7$.

Q: Can the product of the coefficients be 100 or more?

A: Not when you multiply two numbers in scientific notation: both coefficients are less than $10$, so the product is less than $10 \times 10 = 100$. You move the decimal point at most one place. With powers or with three or more factors it can be bigger, so always check.

Q: My calculator shows 1.2E9. What does that mean?

A: Calculators write scientific notation with an E: $1.2\text{E}9$ means $1.2 \times 10^9$. The number after the E is the exponent.