Power Play · Lesson 7 of 7
Chapter Summary and Practise
“Consolidate exponent laws, scientific notation, modelling and mixed chapter practice.”
• Consolidate all exponent laws. • Apply powers to code counts, comparisons and number patterns. • Use scientific notation in mixed problems. • Decide which exponent law fits an expression. • Explain reasoning rather than applying rules mechanically.
Figure it Out
The final problems combine several ideas. Before calculating, identify the structure: same base, same exponent, power of a power, zero or negative exponent, scientific notation, or repeated independent choices.
Problem
Why is 64 both a square and a cube, and what general pattern guarantees both?
- 1.64 = 8² and 64 = 4³.
- 2.A power that is both a square and cube needs an exponent divisible by 2 and 3.
- 3.The smallest such positive exponent is 6.
- 4.Therefore, every sixth power n⁶ is both a square and a cube.
Problem
At least 8.5×10⁹ unique numerical codes are needed. What minimum code length works?
- 1.An n-digit code gives 10ⁿ possibilities.
- 2.10⁹ is too small.
- 3.10¹⁰ is large enough.
- 4.So at least 10 digits are required.
Problem
Which is larger: 2⁸ or 8²?
- 1.2⁸ = 256.
- 2.8² = 64.
- 3.Therefore, 2⁸ is larger.
- 4.Compare the complete values or rewrite them using common bases when possible.
SUMMARY
Power Play begins with repeated doubling and develops a complete language for powers. Exponent notation compresses repeated multiplication. The laws of exponents simplify products and quotients, zero and negative exponents extend the pattern, and powers of 10 support place value, scientific notation and large-number reasoning.
| Situation | Rule |
|---|---|
| Same base, multiply | nᵃ × nᵇ = nᵃ⁺ᵇ |
| Same base, divide | nᵃ ÷ nᵇ = nᵃ⁻ᵇ |
| Power of a power | (nᵃ)ᵇ = nᵃᵇ |
| Same exponent, multiply | mᵃnᵃ = (mn)ᵃ |
| Same exponent, divide | mᵃ/nᵃ = (m/n)ᵃ |
| Zero exponent | n⁰ = 1, n ≠ 0 |
| Negative exponent | n⁻ᵃ = 1/nᵃ, n ≠ 0 |
| Scientific notation | x×10ʸ, 1≤x<10 |
Do not add exponents when bases differ. Do not multiply exponents except in a power of a power. n⁰ is 1 for n≠0. A negative exponent means reciprocal, not a negative value. Scientific notation requires 1≤coefficient<10.
Tremendous in Ten!
The game makes one final point: exponents can produce very large values quickly, but appearances can be misleading. Good number sense means comparing mathematical structure, not merely counting digits or symbols.
Quiz
Simplify 10⁻² × 10⁻⁵.
Simplify (13⁻²)⁻³.
Which is always a square?
A 5-position code has 36 choices per position. How many codes?
Scientific notation for 308,100,000 is:
Practice Problems
- Simplify 5⁷ ÷ 5⁴ and 9⁻⁷ ÷ 9⁴.
- Classify: every cube is a square; every sixth power is both square and cube; product of two cubes is a cube.
- A passcode uses 26 letters and 10 digits with repetition. Find the number of 6-character codes.
- Estimate 8.2×10⁹ people each having 30 objects. Write the total in scientific notation.
- Create two expressions for Tremendous in Ten and justify which is larger.
Key Takeaways
• Exponent laws come from repeated factors. • Same-base multiplication adds exponents; same-base division subtracts them. • A power of a power multiplies exponents. • Zero and negative exponents continue the same division pattern. • Powers of 10 connect place value, scientific notation and large-number sense. • Repeated independent choices naturally create powers. • Strong reasoning means explaining why a rule applies.
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A Pinch of History
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