Exponent Calculator (Powers and Laws of Exponents)

Q: What is a negative exponent?

A negative exponent means “one divided by” the positive power: a⁻ⁿ = 1/aⁿ. So 10⁻² = 1/100 = 0.01 and 2⁻³ = 1/8.

Q: Why is any number to the power 0 equal to 1?

Because aⁿ ÷ aⁿ = 1, and by the quotient law aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. So a⁰ = 1 for every non-zero a. 0⁰ is not defined in school maths.

Q: What are the laws of exponents?

For the same base: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ; and for the same exponent: aⁿ × bⁿ = (ab)ⁿ, aⁿ ÷ bⁿ = (a/b)ⁿ. Also a⁰ = 1 and a⁻ⁿ = 1/aⁿ.

Q: How do I write a big answer in scientific notation?

Move the decimal point so that one non-zero digit is before it, and count the moves: 2¹⁰⁰ = 1267650600228229401496703205376 ≈ 1.2677 × 10³⁰. The calculator shows this automatically.

Q: Can this calculator handle very large powers?

Yes. Answers up to 5,000 digits are worked out exactly with whole-number arithmetic. Larger ones (like 9¹⁰⁰⁰⁰⁰) are given in scientific notation using logarithms.

Type a base and an exponent to work out the power: 2¹⁰ = 1024, 2⁻³ = 1/8, 8^(2/3) = 4. Big answers are exact (2¹⁰⁰ has all 31 digits) and also given in scientific notation. Switch to “Laws of exponents” to simplify aᵐ × aⁿ, aᵐ ÷ aⁿ and (aᵐ)ⁿ the way NCERT Class 7 and 8 teach it.

Base (a)

e.g. 2, −3, 0.5 or 2/3

Exponent (n)

e.g. 10, −3, 1/2 or 2/3

Answer

1024

Decimal

1,024

Scientific notation

1.024 × 10³

Number of digits

Rule used

Repeated multiplication

Step-by-step solution

2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
2¹⁰ = 1,024

How to use the exponent calculator

Choose “Power aⁿ” to evaluate one power, or “Laws of exponents” to combine two powers of the same base.
Type the base: a whole number, a negative number, a decimal (0.5) or a fraction (2/3).
Type the exponent: whole numbers, negative numbers (−3) and fractions (2/3 or 0.5) all work.
Read the exact answer, the decimal and the scientific notation, then the step-by-step working.

Formulas

Product law	`aᵐ × aⁿ = aᵐ⁺ⁿ`
Quotient law	`aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0)`
Power of a power	`(aᵐ)ⁿ = aᵐⁿ`
Zero and negative exponents	`a⁰ = 1, a⁻ⁿ = 1 ÷ aⁿ (a ≠ 0)`
Fractional exponent	`a^(p/q) = (q√a)ᵖ` The denominator is the root, the numerator is the power: 8^(2/3) = (∛8)² = 4.
Scientific notation	`N = m × 10ᵏ, 1 ≤ m < 10`

Solved examples

Example 1: Find 2⁻³.

a⁻ⁿ = 1 ÷ aⁿ
2⁻³ = 1 ÷ 2³ = 1 ÷ 8

Answer: 1/8 = 0.125

Example 2: Simplify 3⁴ × 3² and find its value.

Same base, so add the exponents: 3⁴⁺² = 3⁶
3⁶ = 3 × 3 × 3 × 3 × 3 × 3

Answer: 3⁶ = 729

Example 3: Evaluate 27^(2/3).

a^(p/q) = (q√a)ᵖ
∛27 = 3
3² = 9

Answer: 9

What an exponent means

In aⁿ the base a is multiplied by itself n times: 5³ = 5 × 5 × 5 = 125. The exponent is also called the power or index. Powers of 10 are how we write very large and very small numbers in science: the distance from the Earth to the Sun is about 1.496 × 10¹¹ m.

The laws of exponents only work when the bases are the same. 2³ × 2⁴ = 2⁷, but 2³ × 3⁴ cannot be combined into one power — work out each power separately.

Negative bases and fractional exponents

A negative base raised to an even power is positive, (−2)⁴ = 16, and to an odd power is negative, (−2)³ = −8. Note that −2⁴ means −(2⁴) = −16, so use brackets for a negative base.

A fractional exponent is a root: 16^(1/2) = √16 = 4 and 8^(1/3) = ∛8 = 2. An even root of a negative number, like (−4)^(1/2), is not a real number.

🎯 Asked in competitive exams

This topic regularly appears in the quantitative aptitude section of:

SSC CGL & CHSL
RRB NTPC & Group D
Campus placement aptitude tests

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Frequently asked questions

What is a negative exponent?