Калькулятор степеней

Power rules in brief

A power means repeated multiplication of the same number: a² = a·a and a³ = a·a·a. A negative exponent means the reciprocal, i.e. a⁻¹ = 1/a and a⁻² = 1/a². The exponent 0 always gives the result 1 (when a ≠ 0). A fractional exponent relates to roots: a^(1/2) is the square root and a^(1/3) the cube root.

Как пользоваться калькулятором

1. Enter the base. The number that is raised to a power.
2. Enter the exponent. An integer or a decimal, also negative.
3. Use the shortcuts. x², x³, x⁻¹ and √ (0.5) set the exponent quickly.
4. Calculate. The result and, if needed, scientific notation appear immediately.

Large numbers and scientific notation

As the exponent grows, the result can be very large or very small. In that case the number is presented in scientific form, for example 1.2 × 10⁹. This is the same notation used in physics and engineering to express orders of magnitude. The calculator shows both the ordinary and the scientific form when needed.

Example: compound interest

You invest €1,000 at a return of 5 % a year. After ten years the sum is 1000 · 1.05^10. The power 1.05^10 ≈ 1.6289, so the sum comes to about €1,629. Without compound growth (just 5 % ten times linearly) you would end up at €1,500 — the difference arises precisely because the interest is calculated each year on the already accumulated sum. A power is thus the natural tool for growth and compound interest.

Rules for calculating powers

In the product of powers with the same base, the exponents are added: aᵐ · aⁿ = aᵐ⁺ⁿ. In the quotient they are subtracted: aᵐ / aⁿ = aᵐ⁻ⁿ. In a power of a power, the exponents are multiplied: (aᵐ)ⁿ = aᵐⁿ. With these rules even large expressions simplify quickly, and they explain why a⁰ = 1 (because aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰). The calculator computes the final result, but knowing the rules helps you understand it.

Powers of ten and unit prefixes

Powers are a handy way to express very large and small numbers. The powers of ten give names to unit prefixes: kilo is 10³ (thousand), mega 10⁶ (million), giga 10⁹ (billion) and tera 10¹². In the smaller direction, milli is 10⁻³, micro 10⁻⁶ and nano 10⁻⁹. When you see that 5 GB is 5 × 10⁹ bytes, you immediately grasp the order of magnitude.

In computing, powers of two are also often used, because memory and storage are built on the binary system. 2¹⁰ = 1,024 is close to a thousand, and so the kibibyte (1,024 bytes) and the kilobyte (1,000 bytes) are easily confused. With the power calculator you can compute these orders of magnitude precisely and get a sense of how quickly the values grow when the exponent increases by just a few.

Good to know about powers

• An even power of a negative number is positive, an odd one negative: (−2)² = 4, (−2)³ = −8.
• Any power of one is 1, and a positive power of zero is 0.
• The power of a fraction is calculated for the numerator and denominator separately: (2/3)² = 4/9.
• Powers of ten tell you the order of magnitude quickly, which is handy when comparing large numbers.

Подводя итог

The power calculator raises a number to any exponent, including negative and decimal ones. Enter the base and exponent, and you get the result and, if needed, scientific notation for large numbers. The tool illustrates how quickly values grow as the exponent increases and is suited, for example, to calculating compound interest.

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