“What is 2⁵ × 2³?” — if you hesitated, this article is for you. The five laws of exponents turn repeated multiplication into simple addition and multiplication of the exponents themselves. Once the rules click, power calculations become almost mechanical.
This article introduces all five exponent laws with fully substituted examples, explains negative and zero exponents, and connects them to scientific notation — with a free browser tool for verification.
The five laws
| Law | Formula | Example (base 2) |
|---|---|---|
| Product | bᵐ × bⁿ = bᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | bᵐ ÷ bⁿ = bᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ = 8 |
| Power of a power | (bᵐ)ⁿ = bᵐⁿ | (2²)³ = 2⁶ = 64 |
| Power of a product | (a × b)ⁿ = aⁿ × bⁿ | (2 × 3)² = 36 |
| Zero exponent | b⁰ = 1 | 2⁰ = 1 |
Why each law works
Product law: expanding 2³ × 2⁴ shows three twos followed by four twos — seven in total, hence 2⁷. The exponents count the repetitions, and repetitions add up when multiplied. This law is the foundation; the other four follow from it.
Quotient law: dividing cancels matching factors, so the exponents subtract. Intuitively, “multiplying twice then dividing twice leaves three multiplications.”
Power of a power: (3²)³ repeats the exponent 2 three times, giving 2 × 3 = 6. The power-of-a-product law (ab)ⁿ = aⁿ × bⁿ states that raising a product to a power is the same as raising each factor separately. For example, (2 × 5)³ = 10³ = 1000 equals 2³ × 5³ = 8 × 125 = 1000.
Zero exponent: since 2³ ÷ 2³ = 1 and the quotient law gives 2³⁻³ = 2⁰, it follows that b⁰ = 1.
Negative exponent: since 2² ÷ 2⁵ = 2⁻³ and 2² ÷ 2⁵ = 1/8, a negative exponent means the reciprocal: b⁻ⁿ = 1/bⁿ. The sequence 2⁴ → 2³ → 2² → 2¹ → 2⁰ → 2⁻¹ → 2⁻² reads 16 → 8 → 4 → 2 → 1 → 0.5 → 0.25, halving at each step.
Scientific notation
Scientific notation writes numbers as “a × 10ⁿ” where 1 ≤ a < 10:
- 32000 = 3.2 × 10⁴ (decimal point moved 4 places right)
- 0.001 = 1.0 × 10⁻³ (decimal point moved 3 places left)
Physics and chemistry use this notation as standard because 10ⁿ’s exponent n serves as a magnitude indicator. The speed of light ≈ 3.0 × 10⁸ m/s, Avogadro’s number ≈ 6.02 × 10²³, and the mass of a hydrogen atom ≈ 1.67 × 10⁻²⁷ kg — an enormous range, all unified as “one-digit × 10ⁿ”.
Three common mistakes: multiplying exponents in the product law (bᵐ × bⁿ is bᵐ⁺ⁿ, not bᵐⁿ — multiplication of exponents belongs to the power rule); treating negative exponents as negative numbers (2⁻² is 1/4 = 0.25, not −4); and applying the product law to addition (2² + 2³ = 12, not 2⁵ — the product law only applies to multiplication of the same base).
Verify with the tool
Tools Hub’s Exponent Calculator computes bⁿ with BigInt exact arithmetic for supported integer bases and nonnegative integer exponents, showing the expanded multiplication as working. Negative and fractional exponents are also supported.
How to use it (3 steps)
Enter the base and exponent
Type the base b and exponent n. Integers, negatives and decimals are supported.
Read the result
bⁿ appears in large type — exact BigInt arithmetic for integer exponents, up to 1000 digits.
Check the expansion
The Working panel shows the repeated multiplication, confirming the exponent law being applied.
The tool featured in this article
Exponent Calculator
Base and exponent to bⁿ with BigInt exact arithmetic and expanded multiplication. Negative and fractional exponents supported — free and browser-based.
To reproduce the examples, enter base 2 and exponent 10: the tool shows 1,024 with the expanded multiplication. Enter base 10 and exponent −3 for 0.001. The inverse operation — finding the exponent from the value — is covered in the guide to calculating logarithms. The prime factorization behind integer powers is explained in the prime factorization guide.
Summary
- Five exponent laws: product (add exponents), quotient (subtract), power of a power (multiply), power of a product (distribute), zero exponent (b⁰ = 1)
- Negative exponents are reciprocals: b⁻ⁿ = 1/bⁿ
- Scientific notation is “one-digit × 10ⁿ”, making wide-range numbers comparable
- The product law applies only to multiplication of the same base, not addition
- Verify exponent calculations with a tool that shows its working
FAQ
What is 0⁰?
By convention, 0⁰ = 1 (the empty product). This is consistent with polynomial coefficients and combinatorics, and most textbooks adopt it.
What does a negative exponent mean?
It means taking the reciprocal: b⁻ⁿ = 1/bⁿ. For example, 2⁻³ = 1/8 = 0.125. Flipping the sign of the exponent gives the reciprocal of the result.
What is scientific notation?
A format “a × 10ⁿ” where 1 ≤ a < 10, making wide-range numbers readable. The exponent n indicates the magnitude: 3.2 × 10⁴ = 32000 and 1.0 × 10⁻³ = 0.001.
What happens with fractional exponents?
A fractional exponent like 1/2 corresponds to a root: 2^(1/2) = √2 ≈ 1.414214. Some rational powers of negative bases have real values, such as (−8)^(1/3) = −2, but the tool does not support them through decimal exponent input and reports an error.
References
- OpenStax, College Algebra 2e, Section 1.2 “Exponents and Scientific Notation” (the five exponent laws, negative exponents, and scientific notation): openstax.org
Domains of exponent laws
For integer exponents, exclude zero bases in division and negative powers. General real-exponent laws assume positive bases. The treatment of 0⁰ depends on context; a tool convention is not a universal definition.
