Laws of Exponents
aⁿ means a multiplied by itself n times. Key rules: aᵐ·aⁿ=aᵐ⁺ⁿ, aᵐ/aⁿ=aᵐ⁻ⁿ, (aᵐ)ⁿ=aᵐⁿ. Negative exponent a⁻ⁿ=1/aⁿ. Fractional aᵐ/ⁿ=ⁿ√(aᵐ). Essential for compound interest, growth, scientific notation.
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a⁰=1 for a≠0. a⁻ⁿ=1/aⁿ extends to negative integers. a^(1/n)=ⁿ√a. a^(m/n)=ⁿ√(aᵐ). Scientific notation: N×10ᵏ for 1≤|N|<10.
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Why: Exponents model exponential growth (compound interest, populations), decay (radioactivity), and scientific notation. Moore's law: doubling every 2 years = 2^(t/2).
How: Integer: multiply base repeatedly. Negative: a⁻ⁿ=1/aⁿ. Fractional: aᵐ/ⁿ=ⁿ√(aᵐ). Product rules: add exponents when multiplying same base; subtract when dividing. Power of power: multiply exponents.
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🧮 Fascinating Math Facts
Compound: A=P(1+r/n)^(nt). Doubling time ≈ 72/r%.
— Finance
2^10≈1024, 2^20≈1M. Handy for binary and powers of 2.
— Tech
Key Takeaways
- Product rule: a^m × a^n = a^(m+n) — same base, add exponents.
- Quotient rule: a^m ÷ a^n = a^(m−n) — same base, subtract exponents.
- Power rule: (a^m)^n = a^(m×n) — power of a power, multiply exponents.
- Zero exponent: a^0 = 1 (a ≠ 0).
- Negative exponent: a^(-n) = 1/a^n — reciprocal.
- Fractional exponent: a^(m/n) = ⁿ√(a^m) — roots as fractional powers.
Did You Know?
How It Works
An exponent a^n means multiplying a by itself n times when n is a positive integer. For negative exponents, a^(-n) = 1/a^n. Fractional exponents represent roots: a^(1/n) = ⁿ√a. The laws of exponents (product, quotient, power, zero, negative) allow simplification of expressions. Scientific notation uses powers of 10 to represent very large or small numbers. Logarithms are the inverse: if a^x = b, then x = log_a(b).
Expert Tips
Same base, add exponents
a^m × a^n = a^(m+n). Use when multiplying powers of the same base.
Power of a power
(a^m)^n = a^(mn). Multiply exponents when raising to another power.
Negative = reciprocal
a^(-n) = 1/a^n. Move between numerator and denominator by flipping sign.
Fractional = root
a^(1/n) = ⁿ√a. The nth root is the same as exponent 1/n.
Reference Table — Laws of Exponents
| Rule | Formula |
|---|---|
| Product | a^m × a^n = a^(m+n) |
| Quotient | a^m ÷ a^n = a^(m−n) |
| Power | (a^m)^n = a^(mn) |
| Zero | a^0 = 1 (a≠0) |
| Negative | a^(-n) = 1/a^n |
| Fractional | a^(m/n) = ⁿ√(a^m) |
Frequently Asked Questions
What is a negative exponent?
a^(-n) = 1/a^n. It represents the reciprocal of the positive power.
What is a fractional exponent?
a^(m/n) = ⁿ√(a^m). The denominator is the root, the numerator is the power.
Why is a^0 = 1?
From quotient rule: a^n/a^n = a^(n-n) = a^0, and a^n/a^n = 1.
How does scientific notation use exponents?
Numbers are written as m × 10^n, e.g. 6.02×10²³.
What is the connection to logarithms?
log_a(b) = x means a^x = b. They are inverse operations.
Can you have 0^0?
0^0 is undefined; it arises in limits but has no unique value.
Quick Reference
Disclaimer: Very large or very small results may show as Infinity or 0 due to floating-point limits. Negative base with non-integer exponent yields complex numbers, which are not computed here.
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