AP Chem Score Calculators

AP Chemistry Scientific Notation

Multiplying, dividing, adding, and taking logarithms of numbers in scientific notation, the way chemistry problems use them, in four worked examples.

Chemistry quantities run from 10−34 to 1023, so almost every calculation involves scientific notation. Two features make it more than a math refresher for AP Chemistry: estimating powers of ten in your head is the fastest way to check a calculator answer and catch a slipped exponent, and pH and pK values are logarithms of numbers in scientific notation, with their own significant-figure rule. This page covers both. Counting significant figures in the coefficient is explained on the significant figures page.

The Rules for Calculating in Scientific Notation

To check a conversion or a calculation step by step, enter it in the scientific notation calculator, which also rounds the answer to the right number of significant figures.

The chart places quantities used across the course on one powers-of-ten scale. Their size is why the exponent matters more than the coefficient when you check an answer: a slipped exponent is an error by a factor of 10 or more, while a slipped coefficient is usually a smaller error.

AP Chemistry Quantities on a Powers-of-Ten Scale 10^-30 10^-20 10^-10 10^0 10^10 10^20 Planck constant 6.626 x 10^-34 J s Mass of one H₂O molecule 2.99 x 10^-23 g Energy of a red photon 3.06 x 10^-19 J Kw at 25 C 1.0 x 10^-14 Ksp of AgCl 1.8 x 10^-10 [H+] in pure water 1.0 x 10^-7 M Ka of acetic acid 1.8 x 10^-5 Faraday constant 9.65 x 10^4 C/mol Avogadro’s number 6.022 x 10^23 /mol Labeled ticks are 10 powers of ten apart. Scientific notation keeps these values readable and their significant figures clear.

Worked Example 1: Multiplying and Dividing Chemistry Quantities

(a) How many molecules are in 2.50 × 10−2 mol? (b) For an acid with Ka = 2.5 × 10−5, find Kb of its conjugate base using Kb = Kw/Ka.

(a) (2.50 × 10−2)(6.022 × 1023) = (2.50 × 6.022) × 10−2 + 23 = 15.06 × 1021 = 1.51 × 1022 molecules.
(b) (1.0 × 10−14) ÷ (2.5 × 10−5) = (1.0 ÷ 2.5) × 10−14 − (−5) = 0.40 × 10−9 = 4.0 × 10−10.

In both answers the coefficient landed outside 1 to 10 and had to be shifted: 15.06 becomes 1.506 with the exponent raised by one, and 0.40 becomes 4.0 with the exponent lowered by one. The particle count is the calculation from Avogadro's number, and Kb from Ka is used in salt hydrolysis.

Worked Example 2: Adding Concentrations With Different Powers of Ten

Add 3.2 × 10−4 M and 4.5 × 10−5 M.

Match the powers: 4.5 × 10−5 = 0.45 × 10−4.
Add the coefficients: 3.2 + 0.45 = 3.65, so 3.65 × 10−4 M.
Round by decimal places: 3.2 is known to the tenths place, so the sum is 3.7 × 10−4 M.

Adding coefficients with different exponents directly, 3.2 + 4.5 = 7.7, is the most common error here. A quick check: the smaller term should change the larger one only a little, which it does.

Worked Example 3: Taking Logarithms for pH and Back Again

(a) Find the pH when [H+] = 3.2 × 10−4 M. (b) Find [H+] when the pH is 9.25.

(a) log(3.2 × 10−4) = −4 + log 3.2 = −4 + 0.505 = −3.495, so pH = 3.49. The concentration has two significant figures, so the pH keeps two decimal places.
(b) [H+] = 10−9.25 = 100.75 × 10−10 = 5.6 × 10−10 M, with two significant figures matching the two decimal places.

The part of the pH before the decimal point only records the power of ten, so it does not count as a significant figure. The same rule applies to pOH, pKa, and pKsp. The pH steps themselves are on the strong acids and bases page.

Worked Example 4: Estimating and Taking Square Roots Without a Calculator

(a) Without a calculator, evaluate (3.0 × 10−3)(2.0 × 104) ÷ (6.0 × 102). (b) Find the molar solubility s of AgCl, where Ksp = s2 = 1.8 × 10−10.

(a) Coefficients: 3.0 × 2.0 ÷ 6.0 = 1.0. Exponents: −3 + 4 − 2 = −1. Answer: 1.0 × 10−1.
(b) s = √(1.8 × 10−10). The exponent −10 is even, so s = √1.8 × 10−5 ≈ 1.3 × 10−5 M.

Handling the coefficients and the exponents separately is what makes multiple-choice arithmetic manageable. For a square root with an odd exponent, rewrite first, for example 6.4 × 10−9 as 64 × 10−10, whose root is 8.0 × 10−5. The solubility setup comes from Ksp and molar solubility.

Five Scientific Notation Practice Questions With Answers

  1. Evaluate (4.0 × 10−3)(3.0 × 105).
    Show answer12 × 102 = 1.2 × 103.
  2. Evaluate (8.4 × 10−6) ÷ (2.1 × 10−2).
    Show answer4.0 × 10−6 + 2 = 4.0 × 10−4.
  3. Find the pH when [H+] = 2.5 × 10−11 M.
    Show answer−log(2.5 × 10−11) = 11 − 0.398 = 10.60, with two decimal places for two significant figures.
  4. Find [H+] when the pH is 4.60.
    Show answer10−4.60 = 100.40 × 10−5 = 2.5 × 10−5 M.
  5. Find √(6.4 × 10−9) without a calculator.
    Show answerRewrite as 64 × 10−10; the root is 8.0 × 10−5.

Common Scientific Notation Mistakes

Fluency with powers of ten saves time on every unit, and especially under the time limit of the multiple-choice section. To see how a stronger multiple-choice score would shift your overall estimate, the AP Chem Score Calculator turns section scores into a 1–5 range.

Frequently Asked Questions

How do you multiply numbers in scientific notation?

Multiply the coefficients and add the exponents, then adjust so the coefficient is between 1 and 10. For example, (2.0 × 10³)(4.0 × 10⁵) = 8.0 × 10⁸.

How do you divide numbers in scientific notation?

Divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator. For example, (1.0 × 10⁻¹⁴) ÷ (2.5 × 10⁻⁵) = 0.40 × 10⁻⁹ = 4.0 × 10⁻¹⁰.

How do you add numbers in scientific notation?

Rewrite them with the same power of ten, add the coefficients, then readjust. For example, 3.2 × 10⁻⁴ + 4.5 × 10⁻⁵ = 3.2 × 10⁻⁴ + 0.45 × 10⁻⁴ = 3.65 × 10⁻⁴, reported as 3.7 × 10⁻⁴.

How many decimal places should a pH have?

As many as the significant figures in the concentration. [H⁺] = 3.2 × 10⁻⁴ M has two significant figures, so pH = 3.49 has two decimal places. The digits before the decimal point in a logarithm only show the power of ten.

Can I use a calculator for scientific notation on the AP Chemistry exam?

Yes. College Board's current calculator policy allows a calculator on both the multiple-choice and free-response sections of AP Chemistry, including the built-in Desmos calculator in Bluebook. Estimating powers of ten by hand is still worth practicing, because it catches entry mistakes quickly.

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