Exponent Calculator

Q: What are the laws of exponents?

The six main exponent laws: (1) a^m x a^n = a^(m+n) - multiply same base: add exponents. (2) a^m / a^n = a^(m-n) - divide same base: subtract exponents. (3) (a^m)^n = a^(mn) - power of a power: multiply exponents. (4) (ab)^n = a^n x b^n - power of a product. (5) a^0 = 1 for any a not equal to 0. (6) a^(-n) = 1/a^n. These rules apply to all real exponents.

Q: How do you multiply two numbers with exponents?

When multiplying same bases, add exponents: a^m x a^n = a^(m+n). Example: 2^3 x 2^4 = 2^7 = 128. When dividing, subtract exponents: a^m / a^n = a^(m-n). When raising a power to a power, multiply exponents: (a^m)^n = a^(mn). These are the fundamental laws of exponents used in algebra and scientific notation.

Q: What is the difference between 2^3 and 3^2?

2^3 = 2 x 2 x 2 = 8. 3^2 = 3 x 3 = 9. The base and exponent are not interchangeable - the result is different. In general, a^b does not equal b^a. The only exceptions are when a = b (e.g. 2^2 = 2^2), or special cases like 2^4 = 4^2 = 16. This asymmetry is important to remember when entering values in a calculator.

Calculate x raised to the power of n - including negative and fractional exponents.

What is Exponentiation?

Exponentiation is the mathematical operation of raising a number (the base) to a power (the exponent). Written as xⁿ, it means multiply x by itself n times. For example, 3⁴ = 3 × 3 × 3 × 3 = 81. Exponentiation is one of the five fundamental arithmetic operations alongside addition, subtraction, multiplication, and division.

The concept of exponents is essential across virtually every branch of science and mathematics. In physics, the inverse square law (gravity, light intensity, electric force) all follow power relationships. In finance, compound interest is expressed as P(1 + r)^n. In computer science, algorithm complexity is measured in powers: O(n²) for bubble sort, O(2^n) for brute-force combinatorics. In biology, population growth follows exponential patterns.

Negative exponents represent reciprocals: x^(−n) = 1/xⁿ. This is how very small numbers are expressed in scientific notation. The mass of an electron is approximately 9.11 × 10^(−31) kg. The Planck constant is 6.626 × 10^(−34) J·s. Without negative exponents, working with these numbers would be impractical.

Fractional exponents are another powerful generalization. x^(1/2) means the square root of x, x^(1/3) means the cube root, and in general x^(m/n) means the nth root of x^m. This connects exponentiation directly to roots, showing they are two sides of the same operation. The rules of exponents - product rule, quotient rule, power rule - all follow from the basic definition and apply to all real-number exponents.

Scientific notation expresses numbers as a × 10^b, making it practical to write both astronomically large numbers (distance to the nearest star: ~4.07 × 10^16 metres) and subatomically small ones. This calculator displays results in scientific notation when the value is very large or very small.

Formulas

xⁿ = x × x × x ... (n times)

Product rule: xᵃ × xᵇ = x^(a+b)

Quotient rule: xᵃ ÷ xᵇ = x^(a−b)

Power rule: (xᵃ)ᵇ = x^(ab)

Zero exponent: x⁰ = 1 (x ≠ 0)

Negative exponent: x^(−n) = 1/xⁿ

Fractional exponent: x^(m/n) = (ⁿ√x)^m

How to Use This Calculator

Steps to Calculate an Exponent

Enter the base (x) - any real number. Can be negative, decimal, or a fraction.

Enter the exponent (n) - can be any real number including negative or fractional exponents.

Click Calculate to see the result, scientific notation, common powers of the base (0 through 10), and related values.

Example Calculations

Example 1 - Compound Interest

₹10,000 invested at 8% per annum compounded annually for 20 years

A = P(1 + r)^n = 10,000 × (1.08)^20

(1.08)^20 = 4.6610 (use base 1.08, exponent 20)

A = 10,000 × 4.6610 = ₹46,610

Final amount = ₹46,610

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Example 2 - Negative Exponent

Calculate 4^(−3)

Negative exponent: 4^(−3) = 1/4³ = 1/64

4^(−3) = 0.015625 = 1/64

Try this example →

Example 3 - Fractional Exponent

Calculate 27^(2/3)

27^(2/3) = (∛27)² = 3² = 9

27^(2/3) = 9

Try this example →

Frequently Asked Questions

What is an exponent?+

An exponent (or power) tells you how many times to multiply a number (the base) by itself. For example, 2^5 = 2 × 2 × 2 × 2 × 2 = 32. The base is 2 and the exponent is 5. Exponents are used throughout science, engineering, finance (compound interest), and computing.

What does a negative exponent mean?+

A negative exponent means take the reciprocal. x^(−n) = 1/xⁿ. For example, 5^(−2) = 1/5² = 1/25 = 0.04. Negative exponents appear naturally in scientific notation for very small numbers: 0.001 = 10^(−3).

What does a fractional exponent mean?+

A fractional exponent represents a root. x^(1/n) is the nth root of x. More generally, x^(m/n) = (ⁿ√x)^m or equivalently ⁿ√(x^m). For example, 8^(2/3) = (∛8)² = 2² = 4.

What is x to the power of 0?+

Any non-zero number raised to the power 0 equals 1: x⁰ = 1. This follows from the division rule of exponents: xⁿ/xⁿ = x^(n−n) = x⁰ = 1. The expression 0⁰ is mathematically indeterminate, though in many contexts it is defined as 1 for convenience.

How do you write large numbers in scientific notation?+

Scientific notation expresses a number as a × 10^b, where 1 ≤ a < 10. For example, 299,792,458 (speed of light in m/s) = 2.99792458 × 10⁸. This calculator shows results in scientific notation alongside the standard value for very large or very small numbers.

How are exponents used in compound interest calculations?+

Compound interest relies directly on exponents: A = P × (1 + r/n)^(nt), where n is the compounding frequency per year and t is time in years. The exponent (nt) means the base (1 + r/n) is multiplied by itself nt times. For example, ₹1 lakh at 8% annually for 10 years: A = 1,00,000 × (1.08)^10 = 1,00,000 × 2.1589 = ₹2,15,892. The power of compounding comes entirely from the exponent growing larger with more time.

What is the difference between an exponent and a logarithm?+

Exponentiation and logarithms are inverse operations. If b^x = y, then log_b(y) = x. Example: 2^3 = 8, so log_2(8) = 3. The logarithm answers: "To what power must I raise the base to get this value?" Common logarithms use base 10 (log), natural logarithms use base e ≈ 2.718 (ln). Logarithms are used to solve for the exponent when the result is known - such as finding how many years an investment takes to double.

What are the laws of exponents?+

The six main exponent laws: (1) a^m x a^n = a^(m+n) - multiply same base: add exponents. (2) a^m / a^n = a^(m-n) - divide same base: subtract exponents. (3) (a^m)^n = a^(mn) - power of a power: multiply exponents. (4) (ab)^n = a^n x b^n - power of a product. (5) a^0 = 1 for any a not equal to 0. (6) a^(-n) = 1/a^n. These rules apply to all real exponents.

How do you multiply two numbers with exponents?+

When multiplying same bases, add exponents: a^m x a^n = a^(m+n). Example: 2^3 x 2^4 = 2^7 = 128. When dividing, subtract exponents: a^m / a^n = a^(m-n). When raising a power to a power, multiply exponents: (a^m)^n = a^(mn). These are the fundamental laws of exponents used in algebra and scientific notation.

What is the difference between 2^3 and 3^2?+

2^3 = 2 x 2 x 2 = 8. 3^2 = 3 x 3 = 9. The base and exponent are not interchangeable - the result is different. In general, a^b does not equal b^a. The only exceptions are when a = b (e.g. 2^2 = 2^2), or special cases like 2^4 = 4^2 = 16. This asymmetry is important to remember when entering values in a calculator.

Exponent Calculator

What is Exponentiation?

Formulas

How to Use This Calculator

Steps to Calculate an Exponent

Example Calculations

Example 1 - Compound Interest

₹10,000 invested at 8% per annum compounded annually for 20 years

Example 2 - Negative Exponent

Calculate 4^(−3)

Example 3 - Fractional Exponent

Calculate 27^(2/3)

Frequently Asked Questions

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