Skills Beta

Exponents

An exponent counts repeated multiplication.

Practice 5: Mathematical Routines

Question set for this topic

Part 1 · Hook

Why this matters

A grain of sand holds roughly 50,000,000,000,000,000,000 atoms, and one atom is about 0.0000000001 m across. Writing all those zeros is slow and invites mistakes. Exponents let you write and combine such numbers in a few keystrokes, and they are the main math tool of this course.

Part 2 · Prerequisite check

Quick check before you start

1. Which conversion factor turns milliliters into liters?

  1. 1 L / 1000 mL
  2. 1000 mL / 1 L
  3. 1 L / 100 mL
  4. 100 mL / 1 L
Show the answer

Put the unit you have (mL) on the bottom so it cancels: 1 L / 1000 mL.

  • Correct: 1 L / 1000 mL:
  • 1000 mL / 1 L:
  • 1 L / 100 mL:
  • 100 mL / 1 L:

2. What does the prefix kilo mean?

  1. 1000 times the unit
  2. One thousandth of the unit
  3. 100 times the unit
  4. One millionth of the unit
Show the answer

Kilo multiplies by 1000: 1 km = 1000 m.

  • Correct: 1000 times the unit:
  • One thousandth of the unit:
  • 100 times the unit:
  • One millionth of the unit:

Part 3 · See it

See it first

Four exponent rules: 10² × 10³ = 10⁵ (add), 10⁵ ÷ 10² = 10³ (subtract), (10²)³ = 10⁶ (multiply), and 10⁻³ = 1/1000 = 0.001 (reciprocal).
The four exponent rules, each shown with powers of ten. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

  1. An exponent counts how many times a base is multiplied by itself10³ means 10 × 10 × 10 = 1000
  2. Multiplying two powers of the same base strings their factors togethertheir exponents add
  3. Dividing cancels factors that appear on the top and the bottomthe exponents subtract, and a larger bottom exponent leaves a negative exponent
  4. A negative exponent means factors left on the bottom10⁻ⁿ is 1/10ⁿ, a small positive number

Part 5 · Key ideas

Key ideas

  • An exponent says how many times to multiply the base by itself: 10⁴ = 10,000.
  • Multiply: add exponents (10² × 10³ = 10⁵). Divide: subtract (10² ÷ 10⁵ = 10⁻³). Power of a power: multiply (10²)³ = 10⁶.
  • Negative exponent = reciprocal: 10⁻³ = 1/1000 = 0.001. Zero exponent: 10⁰ = 1.
  • Roots are fractional powers: √(10⁸) = 10⁴. Adding numbers does not combine exponents: 10⁴ + 10⁴ = 2 × 10⁴.

Part 6 · Misconception

A common mistake

The wrong idea: A negative exponent makes the number negative, so 10⁻³ = −1000.

What actually happens: A negative exponent means divide: 10⁻³ = 1/10³ = 0.001, a small positive number. The sign of the number itself comes from a minus sign in front, not from the exponent.

Part 7 · Check yourself

Check yourself

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

1. What is 10³ × 10⁴?

  1. 10¹²
  2. 100⁷
  3. 10⁷
  4. 10¹
Show the answer

10³ × 10⁴ = (10 × 10 × 10) × (10 × 10 × 10 × 10) = 10⁷: seven tens multiplied together.

  • 10¹²: This multiplies the exponents. When you multiply powers of the same base, you add them.
  • 100⁷: The base stays 10; only the exponent changes.
  • Correct: 10⁷: Right: same base, so add the exponents: 3 + 4 = 7.
  • 10¹: This subtracts the exponents, which is the rule for dividing.

2. What is 10⁵ ÷ 10⁸?

  1. 10⁻³
  2. 10³
  3. 10¹³
  4. 10⁴⁰
Show the answer

Dividing powers of the same base subtracts the exponents: 10⁵⁻⁸ = 10⁻³ = 0.001.

  • Correct: 10⁻³: Right: subtract the exponents, 5 − 8 = −3.
  • 10³: The sign is backwards: the bottom exponent is larger, so the result is less than 1.
  • 10¹³: This adds the exponents, the rule for multiplying, not dividing.
  • 10⁴⁰: This multiplies the exponents, which is the rule for raising a power to a power.

3. What is (10⁻²)³?

  1. 10¹
  2. 10⁻⁸
  3. 10⁶
  4. 10⁻⁶
Show the answer

(10⁻²)³ = 10⁻² × 10⁻² × 10⁻² = 10⁻⁶.

  • 10¹: This adds −2 and 3. Raising a power to a power multiplies the exponents.
  • 10⁻⁸: This raises 2 to the third power. The rule is to multiply the exponents.
  • 10⁶: The size is right but the sign is lost: −2 × 3 is −6.
  • Correct: 10⁻⁶: Right: a power of a power multiplies the exponents, −2 × 3 = −6.

4. What does a negative exponent mean, as in 10⁻⁴?

  1. A negative number, −10,000
  2. One divided by 10⁴, which is 0.0001
  3. 10 multiplied by −4, which is −40
  4. The number 10 with its digits reversed, four times
Show the answer

10⁻⁴ = 1 / 10⁴ = 1/10,000 = 0.0001. A negative exponent gives a small positive number.

  • A negative number, −10,000: A negative exponent never makes the number negative; 10⁻⁴ is a small positive number.
  • Correct: One divided by 10⁴, which is 0.0001: Right: a negative exponent means the reciprocal: 1/10⁴.
  • 10 multiplied by −4, which is −40: An exponent counts how many times the base is multiplied, not a number to multiply by.
  • The number 10 with its digits reversed, four times: Exponents are about repeated multiplication, not digit order.

5. Write 10⁻³ as an ordinary decimal number.

Type a number.

Show the answer

10⁻³ = 1 / 10³ = 1/1000 = 0.001. Count the places: the 1 lands in the third place after the decimal point.

  • Answer: 0.001

6. A student writes 10⁴ + 10⁴ = 10⁸. What is the error?

  1. The exponent rules apply to multiplying, not adding; the sum is 20,000
  2. The exponents should have been multiplied to give 10¹⁶
  3. The answer should be 10⁴, because the exponents are equal
  4. The student made no error; adding exponents is the rule here
Show the answer

10⁴ + 10⁴ = 2 × 10⁴ = 20,000. Exponent rules only combine exponents in multiplication and division.

  • Correct: The exponent rules apply to multiplying, not adding; the sum is 20,000: Right: 10,000 + 10,000 = 20,000, which is 2 × 10⁴, not 10⁸.
  • The exponents should have been multiplied to give 10¹⁶: Multiplying exponents is for a power raised to a power, and this is a sum.
  • The answer should be 10⁴, because the exponents are equal: 10⁴ + 10⁴ is twice 10⁴, not 10⁴ itself.
  • The student made no error; adding exponents is the rule here: Adding exponents is the rule for multiplying powers, and this problem adds them.

7. Evaluate (10⁻⁴ × 10⁷) ÷ 10². Give the answer as an ordinary number.

Type a number.

Show the answer

Top: 10⁻⁴⁺⁷ = 10³. Then 10³ ÷ 10² = 10¹ = 10.

  • Answer: 10

Part 8 · Summary

Summary

An exponent counts repeated multiplication. Multiplying powers of the same base adds exponents, dividing subtracts them, and raising a power to a power multiplies them. A negative exponent means a reciprocal, and any nonzero base to the zero power is 1. These rules do not apply to adding.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections