Scientific Notation Operations

Scientific notation is a way of writing very large or very small numbers in a compact form. Instead of writing 93,000,000 miles (the distance from Earth to the Sun) or 0.000001 grams (the mass of a tiny particle), you write $9.3 \times 10^7$ and $1 \times 10^{-6}$. The format is always:

$a \times 10^n$

where $1 \leq a < 10$ (a number with exactly one nonzero digit before the decimal point) and $n$ is an integer exponent. This page covers how to perform arithmetic — multiplication, division, addition, and subtraction — with numbers already in scientific notation.

Review: Converting to Scientific Notation

Before diving into operations, a quick refresher on the format:

• Large numbers: Move the decimal left until you have one digit before it. The exponent is positive and equals the number of places you moved. Example: $45{,}000 = 4.5 \times 10^4$.
• Small numbers: Move the decimal right until you have one digit before it. The exponent is negative. Example: $0.0032 = 3.2 \times 10^{-3}$.

Multiplying in Scientific Notation

To multiply two numbers in scientific notation, multiply the coefficients and add the exponents:

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

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

Example 1: $(3 \times 10^4)(2 \times 10^5)$

Step 1 — Multiply the coefficients:

$3 \times 2 = 6$

Step 2 — Add the exponents:

$10^4 \times 10^5 = 10^{4+5} = 10^9$

Step 3 — Combine:

$6 \times 10^9$

The coefficient 6 is between 1 and 10, so no adjustment needed.

Answer: $6 \times 10^9$

Example 2: $(4.5 \times 10^3)(6 \times 10^7)$

Step 1 — Multiply the coefficients:

$4.5 \times 6 = 27$

Step 2 — Add the exponents:

$10^3 \times 10^7 = 10^{10}$

Step 3 — Combine:

$27 \times 10^{10}$

Step 4 — Adjust: 27 is not between 1 and 10. Rewrite as $2.7 \times 10^1$, then:

$2.7 \times 10^1 \times 10^{10} = 2.7 \times 10^{11}$

Answer: $2.7 \times 10^{11}$

Example 3: $(8 \times 10^{-3})(5 \times 10^{-4})$

Step 1 — Multiply coefficients: $8 \times 5 = 40$

Step 2 — Add exponents: $10^{-3} \times 10^{-4} = 10^{-7}$

Step 3 — Combine: $40 \times 10^{-7}$

Step 4 — Adjust: $40 = 4.0 \times 10^1$, so:

$4.0 \times 10^1 \times 10^{-7} = 4.0 \times 10^{-6}$

Answer: $4.0 \times 10^{-6}$

Dividing in Scientific Notation

To divide, divide the coefficients and subtract the exponents:

$\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$

Example 4: $\dfrac{8.4 \times 10^9}{2.1 \times 10^3}$

Step 1 — Divide the coefficients:

$\frac{8.4}{2.1} = 4$

Step 2 — Subtract the exponents:

$10^{9-3} = 10^6$

Step 3 — Combine:

$4 \times 10^6$

Answer: $4 \times 10^6$

Example 5: $\dfrac{3.6 \times 10^{-2}}{9 \times 10^{4}}$

Step 1 — Divide coefficients:

$\frac{3.6}{9} = 0.4$

Step 2 — Subtract exponents:

$10^{-2-4} = 10^{-6}$

Step 3 — Combine:

$0.4 \times 10^{-6}$

Step 4 — Adjust: $0.4 = 4 \times 10^{-1}$, so:

$4 \times 10^{-1} \times 10^{-6} = 4 \times 10^{-7}$

Answer: $4 \times 10^{-7}$

Adding and Subtracting in Scientific Notation

Addition and subtraction require an extra step: the exponents must be the same before you can combine the coefficients. You cannot simply add exponents like you do when multiplying.

Strategy: Rewrite one (or both) numbers so that both have the same power of 10, then add or subtract the coefficients.

Example 6: $(5.2 \times 10^4) + (3.1 \times 10^4)$

The exponents already match. Add the coefficients:

$5.2 + 3.1 = 8.3$

Answer: $8.3 \times 10^4$

Example 7: $(6.4 \times 10^5) + (8.0 \times 10^3)$

The exponents differ. Rewrite $8.0 \times 10^3$ with an exponent of 5:

$8.0 \times 10^3 = 0.080 \times 10^5$

Now add:

$6.4 \times 10^5 + 0.080 \times 10^5 = (6.4 + 0.080) \times 10^5 = 6.48 \times 10^5$

Answer: $6.48 \times 10^5$

Example 8: $(9.1 \times 10^{-2}) - (4.3 \times 10^{-3})$

Rewrite $4.3 \times 10^{-3}$ with an exponent of $-2$:

$4.3 \times 10^{-3} = 0.43 \times 10^{-2}$

Subtract:

$9.1 \times 10^{-2} - 0.43 \times 10^{-2} = (9.1 - 0.43) \times 10^{-2} = 8.67 \times 10^{-2}$

Answer: $8.67 \times 10^{-2}$

Real-World Application: Nursing — Blood Cell Counts

A complete blood count (CBC) reports values in scientific notation. A healthy adult might have approximately $5.0 \times 10^6$ red blood cells per microliter (often written as $5.0 \times 10^6 / \mu\text{L}$) and $7.5 \times 10^3$ white blood cells per microliter.

Scenario: A nurse needs to determine how many times more red blood cells there are than white blood cells.

$\frac{5.0 \times 10^6}{7.5 \times 10^3}$

Step 1 — Divide coefficients:

$\frac{5.0}{7.5} = 0.667$

Step 2 — Subtract exponents:

$10^{6-3} = 10^3$

Step 3 — Combine and adjust:

$0.667 \times 10^3 = 6.67 \times 10^2 = 667$

Answer: There are approximately 667 times more red blood cells than white blood cells in a microliter of blood. Scientific notation makes this comparison manageable despite the vastly different magnitudes.

Real-World Application: Electrician — Wire Resistance Calculations

Electricians work with very small resistance values. Copper wire has a resistivity of approximately $1.7 \times 10^{-8}$ ohm-meters. For a wire with cross-sectional area $3.31 \times 10^{-6}$ square meters and length 50 meters:

$R = \frac{\rho \times L}{A} = \frac{(1.7 \times 10^{-8})(50)}{3.31 \times 10^{-6}}$

Step 1 — Multiply numerator: $(1.7 \times 10^{-8})(5 \times 10^1) = 8.5 \times 10^{-7}$

Step 2 — Divide: $\dfrac{8.5 \times 10^{-7}}{3.31 \times 10^{-6}} = \dfrac{8.5}{3.31} \times 10^{-7-(-6)} = 2.57 \times 10^{-1} = 0.257$ ohms

Answer: The wire has a resistance of approximately 0.257 ohms. Keeping values in scientific notation prevents errors with all those decimal places.

Common Mistakes to Avoid

1. Adding exponents when adding numbers. $(3 \times 10^4) + (2 \times 10^4) = 5 \times 10^4$, not $5 \times 10^8$. You add exponents only when multiplying.
2. Forgetting to adjust the coefficient. After multiplying, if the coefficient is 27 (not between 1 and 10), you must rewrite it as $2.7 \times 10^1$ and adjust the exponent.
3. Subtracting exponents in the wrong order. In division, subtract the denominator’s exponent from the numerator’s: $10^{m-n}$. Getting it backward gives the wrong sign.
4. Adding numbers with different exponents without converting first. You must match the exponents before adding or subtracting coefficients.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multiplying: Multiply the coefficients and add the exponents — $(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$
• Dividing: Divide the coefficients and subtract the exponents — $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$
• Adding/Subtracting: First make the exponents the same, then combine the coefficients
• Always adjust the final answer so the coefficient is between 1 and 10
• Scientific notation is essential in healthcare (cell counts, dosages) and electrical work (resistivity, current) where values span many orders of magnitude

Return to Algebra for more topics in this section.