What are the rules of exponents with examples?

The 8 rules are: (1) Product Rule a^m times a^n equals a^(m+n) e.g. 2^3 times 2^4 equals 2^7 equals 128, (2) Quotient Rule a^m divided by a^n equals a^(m-n), (3) Power Rule (a^m)^n equals a^(mn), (4) Power of Product (ab)^n equals a^n times b^n, (5) Power of Quotient (a/b)^n equals a^n/b^n, (6) Negative Exponent a^(-n) equals 1/a^n, (7) Zero Exponent a^0 equals 1, (8) Fractional Exponent a^(m/n) equals nth root of a^m. Enter any values to see each rule applied step by step.

How do you solve negative exponents step by step?

Step 1 write the base with the positive exponent in the denominator. Step 2 calculate the positive power. Step 3 take the reciprocal. Example: 5^(-3) equals 1/5^3 equals 1/125 equals 0.008. The general rule is a^(-n) equals 1/a^n. This works because multiplying a^n times a^(-n) must equal a^0 which equals 1, so a^(-n) equals 1/a^n.

How do you simplify expressions with exponents?

To simplify exponent expressions: first apply the power rule to nested powers like (a^2)^3 equals a^6, then use the product rule to combine same-base terms like a^6 times a^4 equals a^10, then use the quotient rule for division. For example (x^3)^2 times x^4 divided by x^5 simplifies to x^6 times x^4 divided by x^5 equals x^10 divided by x^5 equals x^5. This calculator shows every intermediate step.

Why is anything to the power of 0 equal to 1?

By the quotient rule a^n divided by a^n equals a^(n minus n) equals a^0. But any number divided by itself is 1. Therefore a^0 must equal 1. Another way to see it: each time you decrease the exponent by 1 you divide by the base. Starting from 2^3 equals 8, 2^2 equals 4, 2^1 equals 2, so 2^0 equals 1. This holds for every non-zero base. Note 0^0 is undefined.

What is a fractional or rational exponent?

A fractional exponent a^(m/n) means take the nth root of a raised to the m power. The denominator n is the root index and the numerator m is the power. Example: 27^(2/3) means cube root of 27 squared. Cube root of 27 is 3, then 3 squared is 9. You can also compute 27^2 first to get 729, then take the cube root of 729 which also gives 9. Both methods always yield the same answer.

Exponent Calculator with Steps - All 8 Laws of Exponents

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What are Exponents?

An exponent tells you how many times to multiply a number (the base) by itself. For example, 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. Exponents are fundamental in algebra, physics, computer science, and finance — appearing everywhere from compound interest to binary numbers.

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Key Insight: The 8 laws of exponents let you simplify complex expressions by combining, splitting, and transforming powers — without ever computing the full result. This is essential for algebra and calculus.

The 8 Laws of Exponents

These rules govern how exponents behave in multiplication, division, and nesting.

1. Product Rule

a^m × aⁿ = a^m+n

Same base? Add the exponents.

2. Quotient Rule

a^m ÷ aⁿ = a^m−n

Same base? Subtract the exponents.

3. Power Rule

(a^m)ⁿ = a^mn

Power of a power? Multiply.

4. Power of Product

(ab)ⁿ = aⁿbⁿ

Distribute across multiplication.

5. Power of Quotient

(a/b)ⁿ = aⁿ/bⁿ

Distribute across division.

6. Negative Exponent

a⁻ⁿ = 1/aⁿ

Flip to the denominator.

7. Zero Exponent

a⁰ = 1

Anything (non-zero) to the zero = 1.

8. Fractional Exponent

a^m/n = ⁿ√(a^m)

Numerator = power, denominator = root.

Real-World Applications

Exponents appear throughout mathematics, science, and everyday life.

Compound Interest

A = P(1 + r/n)^nt. Your savings grow exponentially thanks to the power of compounding.

Computer Science

Binary uses powers of 2. A 32-bit integer stores values up to 2³² − 1 = 4,294,967,295.

Physics & Engineering

Radioactive decay (N = N₀ · 2^−t/h), sound intensity (decibels), and earthquake magnitudes all use exponents.

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Try it! Enter base = 1.08, exponent = 30 to see how $1 grows to $10.06 at 8% annual compound interest over 30 years.

Frequently Asked Questions

The 8 laws are: (1) Product Rule: a^m × aⁿ = a^m+n, (2) Quotient Rule: a^m ÷ aⁿ = a^m−n, (3) Power Rule: (a^m)ⁿ = a^mn, (4) Power of a Product: (ab)ⁿ = aⁿbⁿ, (5) Power of a Quotient: (a/b)ⁿ = aⁿ/bⁿ, (6) Negative Exponent: a⁻ⁿ = 1/aⁿ, (7) Zero Exponent: a⁰ = 1, (8) Fractional Exponent: a^m/n = ⁿ√(a^m).

A negative exponent means take the reciprocal: a⁻ⁿ = 1/aⁿ. For example, 2⁻³ = 1/2³ = 1/8 = 0.125. First compute the positive power, then take the reciprocal.

A fractional exponent combines a power and a root. The numerator is the power and the denominator is the root. For example, 8^2/3 means the cube root of 8 squared. Cube root of 8 is 2, then 2 squared is 4. So 8^2/3 = 4.

Using the quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But any number divided by itself equals 1, so a⁰ = 1. This holds for any non-zero base. Note that 0⁰ is typically considered undefined.

Yes, 100% free with no signup. Features include all 8 exponent laws with step-by-step solutions, multi-rule simplification, an All Laws comparison mode, a Python SymPy compiler, LaTeX export, and shareable URLs. All computation runs in your browser.

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What are Exponents?

The 8 Laws of Exponents

1. Product Rule

2. Quotient Rule

3. Power Rule

4. Power of Product

5. Power of Quotient

6. Negative Exponent

7. Zero Exponent

8. Fractional Exponent

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What are Exponents?

The 8 Laws of Exponents

1. Product Rule

2. Quotient Rule

3. Power Rule

4. Power of Product

5. Power of Quotient

6. Negative Exponent

7. Zero Exponent

8. Fractional Exponent

Real-World Applications

Compound Interest

Computer Science

Physics & Engineering

Frequently Asked Questions

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Logarithm Calculator

Quadratic Solver

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