Number & operations · Grade 8 · Study notes
Exponents, Powers and Scientific Notation
Exponents are a compact way to write repeated multiplication. They also help describe very large and very small quantities.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Interpret powers as repeated multiplication and evaluate powers with positive, zero and negative integer exponents.
- Apply the product, quotient and power-of-a-power laws when the bases are the same.
- Write very large and very small values in scientific notation and convert them back to ordinary form.
- Check that the coefficient in scientific notation has an absolute value from 1 up to, but not including, 10.
1. What an exponent means
In 4³, the base is 4 and the exponent is 3: 4 × 4 × 4 = 64. The exponent tells how many times the base is used as a factor. It is not multiplication by the exponent.
2. Zero and negative exponents
For any non-zero number a, a⁰ = 1. A negative exponent means reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/8. Zero to a negative power is undefined.
3. Laws of exponents
For the same non-zero base: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0); (aᵐ)ⁿ = aᵐⁿ. Also (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ when defined. Add exponents only when multiplying like bases.
4. Powers of ten
10⁰ = 1, 10¹ = 10, 10² = 100, and 10⁻² = 0.01. Each step up in a positive exponent multiplies by 10; each step down divides by 10.
5. Scientific notation
Write a number as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. For example, 4,500,000 = 4.5 × 10⁶ and 0.00072 = 7.2 × 10⁻⁴. Move the decimal left for a large number (positive exponent) and right for a small number (negative exponent).
Apply exponent laws carefully
For the same non-zero base, multiply powers by adding exponents: am × an = am+n. Divide powers by subtracting exponents, and raise a power to another power by multiplying the exponents. A zero exponent gives 1 for a non-zero base. A negative exponent means take the reciprocal: 10−2 = 1/100 = 0.01.
Worked reasoning: 23 × 24 = 27 = 128. But 23 + 24 = 8 + 16 = 24; the product rule does not apply to addition.
In scientific notation, the first factor must have an absolute value at least 1 but less than 10. For example, 0.00056 = 5.6 × 10−4.
- Keep brackets when a negative number is the base: (−2)2 = 4, while −22 = −4.
- Do not add exponents when the bases are different.
- Check that the decimal point moves in the correct direction.
Worked examples
- 3⁴ = 3 × 3 × 3 × 3 = 81.
- 5² × 5³ = 5⁵ = 3,125.
- (2³)² = 2⁶ = 64.
- 6.2 × 10⁵ = 620,000; 3.4 × 10⁻³ = 0.0034.
Independent practice
Try each question before revealing or checking the worked answer.
- Evaluate 2⁵. Answer: 32.
- Evaluate 10⁻³. Answer: 0.001.
- Simplify x⁴ × x². Answer: x⁶.
- Simplify (3²)³. Answer: 3⁶ = 729.
- Write 8,100,000 in scientific notation. Answer: 8.1 × 10⁶.
- Write 0.000056 in scientific notation. Answer: 5.6 × 10⁻⁵.
- Simplify y⁷ ÷ y³ (y ≠ 0). Answer: y⁴.
- Evaluate 4⁰ + 2³. Answer: 1 + 8 = 9.
Knowledge check
Choose one answer for each question, then check your answers. Explanations appear after checking.
Printable worksheet
Use the independent-practice questions above as your worksheet. Show each step, include units where relevant, and check answers only after attempting the questions.
Answer key
- Evaluate 2⁵. Answer: 32.
- Evaluate 10⁻³. Answer: 0.001.
- Simplify x⁴ × x². Answer: x⁶.
- Simplify (3²)³. Answer: 3⁶ = 729.
- Write 8,100,000 in scientific notation. Answer: 8.1 × 10⁶.
- Write 0.000056 in scientific notation. Answer: 5.6 × 10⁻⁵.
- Simplify y⁷ ÷ y³ (y ≠ 0). Answer: y⁴.
- Evaluate 4⁰ + 2³. Answer: 1 + 8 = 9.
Study tip: Write down the rule you used beside each answer. This makes errors easier to find and correct.