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Scientific Notation

Scientific notation writes a number as a coefficient from 1 to just under 10 times a power of ten.

Practice 5: Mathematical Routines

Question set for this topic

Part 1 · Hook

Why this matters

The mass of one hydrogen atom is 0.000000000000000000000001674 g. Count those zeros twice and you may still get it wrong. Scientists write it as 1.674 × 10⁻²⁴ g instead: shorter, easier to compare, and easier to type into a calculator without slipping.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is 10⁴ × 10⁻⁷?

  1. 10⁻³
  2. 10¹¹
  3. 10⁻²⁸
  4. 10³
Show the answer

Multiplying powers of ten adds the exponents: 4 + (−7) = −3.

  • Correct: 10⁻³:
  • 10¹¹:
  • 10⁻²⁸:
  • 10³:

2. What is 10⁻² as a decimal?

  1. 0.01
  2. −100
  3. 0.1
  4. −0.01
Show the answer

A negative exponent is a reciprocal: 1/10² = 1/100 = 0.01.

  • Correct: 0.01:
  • −100:
  • 0.1:
  • −0.01:

Part 4 · See it

See it first

149,600,000 becomes 1.496 × 10⁸ (point moved 8 places left) and 0.000000650 becomes 6.50 × 10⁻⁷ (point moved 7 places right); the coefficient is between 1 and 10.
Scientific notation: a coefficient from 1 to just under 10, times a power of ten that says how many places the decimal point moved. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Very large and very small numbers carry many zerosthey are slow to write and easy to miscount
  2. Any number can be split into a coefficient between 1 and 10 and a power of tenthe zeros move into the exponent
  3. Moving the decimal point left makes the coefficient smallerthe exponent goes up by one for each place, so a large number gets a positive exponent
  4. The exponent shows the size at a glancecomparing, multiplying and dividing become exponent arithmetic

Part 6 · Key ideas

Key ideas

  • Scientific notation: coefficient × 10ⁿ, with 1 ≤ coefficient < 10. 0.00560 = 5.60 × 10⁻³; 149,600,000 = 1.496 × 10⁸.
  • Numbers above 10 get positive exponents; numbers below 1 get negative exponents.
  • Multiply: multiply coefficients, add exponents. Divide: divide coefficients, subtract exponents. Then fix the coefficient if it left the 1-10 range.
  • Add or subtract only after giving both numbers the same exponent. One order of magnitude is a factor of 10.

Part 7 · Misconception

A common mistake

The wrong idea: A larger coefficient always means a larger number, so 8 × 10⁻⁴ is bigger than 3 × 10⁻².

What actually happens: Compare the exponents first. 8 × 10⁻⁴ = 0.0008 and 3 × 10⁻² = 0.03, so the second is about 40 times larger. The coefficient only decides between numbers with the same exponent.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. Which number is written in correct scientific notation?

  1. 47 × 10⁻⁴
  2. 0.47 × 10⁻²
  3. 4.7 × 10⁻³
  4. 4.7 × 3⁻³
Show the answer

In scientific notation the coefficient is at least 1 and less than 10: 4.7 × 10⁻³ = 0.0047.

  • 47 × 10⁻⁴: Same value, but the coefficient 47 is 10 or more, so it is not in standard form.
  • 0.47 × 10⁻²: Same value, but the coefficient 0.47 is less than 1.
  • Correct: 4.7 × 10⁻³: Right: the coefficient is between 1 and 10, times a power of ten.
  • 4.7 × 3⁻³: Scientific notation uses powers of ten, not powers of 3.

2. A fine smoke particle is 0.000000650 m across. Write it in scientific notation (type it as, for example, 6.50e-7) in meters.

Type a number in m.

Show the answer

Move the decimal point right until one nonzero digit is in front: 7 places, so the exponent is −7. 6.50 × 10⁻⁷ m. Moving right for a small number gives a negative exponent.

  • Answer: 6.50 × 10-7 m

3. Calculate (3.0 × 10⁴) × (2.0 × 10⁻⁶).

Type a number.

Show the answer

Multiply the coefficients: 3.0 × 2.0 = 6.0. Add the exponents: 4 + (−6) = −2. The answer is 6.0 × 10⁻².

  • Answer: 0.060

4. Calculate (8.4 × 10⁶) ÷ (2.1 × 10⁻³).

Type a number.

Show the answer

Divide the coefficients: 8.4 ÷ 2.1 = 4.0. Subtract the exponents: 6 − (−3) = 9. The answer is 4.0 × 10⁹.

  • Answer: 4.0 × 109

5. A calculation gives 52 × 10⁵. What is this in correct scientific notation?

  1. 5.2 × 10⁴
  2. 52 × 10⁵
  3. 0.52 × 10⁷
  4. 5.2 × 10⁶
Show the answer

52 × 10⁵ = 5.2 × 10¹ × 10⁵ = 5.2 × 10⁶. When the coefficient shrinks, the exponent grows to keep the value the same.

  • 5.2 × 10⁴: The coefficient got 10 times smaller, so the exponent must go up by one, not down.
  • 52 × 10⁵: The value is right, but a coefficient of 52 is not between 1 and 10.
  • 0.52 × 10⁷: The value is right, but a coefficient of 0.52 is less than 1.
  • Correct: 5.2 × 10⁶: Right: dividing the coefficient by 10 means multiplying the power of ten by 10.

6. Which list is in order from smallest to largest?

  1. 8 × 10⁻⁴, 3 × 10⁻⁵, 2 × 10⁻², 1 × 10³
  2. 1 × 10³, 2 × 10⁻², 8 × 10⁻⁴, 3 × 10⁻⁵
  3. 3 × 10⁻⁵, 8 × 10⁻⁴, 2 × 10⁻², 1 × 10³
  4. 2 × 10⁻², 8 × 10⁻⁴, 3 × 10⁻⁵, 1 × 10³
Show the answer

3 × 10⁻⁵ = 0.00003, 8 × 10⁻⁴ = 0.0008, 2 × 10⁻² = 0.02 and 1 × 10³ = 1000. With the coefficient between 1 and 10, the exponent decides the order.

  • 8 × 10⁻⁴, 3 × 10⁻⁵, 2 × 10⁻², 1 × 10³: This sorts by coefficient first. A larger coefficient does not beat a more negative exponent: 0.0008 is more than 0.00003.
  • 1 × 10³, 2 × 10⁻², 8 × 10⁻⁴, 3 × 10⁻⁵: This is largest to smallest, the reverse order.
  • Correct: 3 × 10⁻⁵, 8 × 10⁻⁴, 2 × 10⁻², 1 × 10³: Right: compare exponents first; the most negative exponent is the smallest number.
  • 2 × 10⁻², 8 × 10⁻⁴, 3 × 10⁻⁵, 1 × 10³: This treats −2 as smaller than −5. On the number line, 10⁻⁵ is far smaller than 10⁻².

7. A sample contains 1.5 × 10²¹ molecules, and each molecule has a mass of 3.0 × 10⁻²³ g. What is the total mass in grams?

Type a number in g.

Show the answer

(1.5 × 10²¹) × (3.0 × 10⁻²³ g) = 4.5 × 10⁻² g = 0.045 g. Coefficients multiply (1.5 × 3.0 = 4.5) and exponents add (21 − 23 = −2).

  • Answer: 0.045 g

Part 9 · Summary

Summary

Scientific notation writes a number as a coefficient from 1 to just under 10 times a power of ten. Large numbers have positive exponents and small numbers negative ones. To multiply, multiply coefficients and add exponents; to divide, divide and subtract; to add, match exponents first. Each factor of ten is one order of magnitude.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections