Powers & Roots Calculator - Exponents, Square Root, Cube Root

    Raise to any power, extract square roots (√), cube roots (∛) and nth roots. Supports decimal and negative exponents.

    Parameters

    Enter data for calculations

    Choose the mathematical operation: exponentiation (a^n), square root (√), cube root (∛) or nth root

    The base number - for power it is the base (a in a^n), for root it is the radicand (a in √a)

    Power exponent (n in a^n) or root degree (n in ⁿ√a). Leave empty for square and cube root

    Form progress0 / 2 fields

    💡 Fill in all required fields to unlock the calculate button

    How Does the Powers & Roots Calculator Work?

    Select an operation (power, square root, cube root, or nth root), enter the base number, and optionally the exponent or degree. The calculator returns the result with the mathematical formula and a plain-language explanation. It handles decimal exponents, negative bases (for odd roots), and zero exponent - all edge cases that trip people up when doing these by hand.

    Formulas and Rules

    Operation Notation Example Result
    Power a^n 2^10 1024
    Square root √a √144 12
    Cube root ∛a ∛27 3
    Nth root n√a 4√625 5

    Key Rules of Exponents

    Rule Formula Example
    Product of powers a^m x a^n = a^(m+n) 2^3 x 2^4 = 2^7 = 128
    Power of a power (a^m)^n = a^(m x n) (2^3)^2 = 2^6 = 64
    Zero exponent a^0 = 1 (a not 0) 5^0 = 1
    Negative exponent a^(-n) = 1 / a^n 2^(-3) = 1/8 = 0.125
    Fractional exponent a^(1/n) = n√a 8^(1/3) = ∛8 = 2

    Common Powers Reference

    n 2^n 3^n 5^n 10^n
    1 2 3 5 10
    2 4 9 25 100
    3 8 27 125 1 000
    5 32 243 3 125 100 000
    10 1 024 59 049 9 765 625 10 000 000 000

    Practical Examples

    Example 1: 5^3 (5 cubed)
    Result: 125 (5 x 5 x 5)
    Example 2: √256 (square root of 256)
    Result: 16 (16 x 16 = 256)
    Example 3: ∛1000 (cube root of 1000)
    Result: 10 (10 x 10 x 10 = 1000)
    Example 4: Compound interest - 1000 x (1.05)^10
    Result: 1628.89 (1000 dollars at 5% annual for 10 years)
    Example 5: Binary: 2^16 (16-bit range)
    Result: 65 536 (max value for unsigned 16-bit integer)
    Example 6: 5th root of 32 (5√32)
    Result: 2 (2^5 = 32)
    Example 7: Negative exponent: 10^(-3)
    Result: 0.001 (1/1000 - used in metric prefixes: milli = 10^-3)

    FAQ - Frequently Asked Questions

    What is exponentiation?
    Exponentiation is repeated multiplication. a^n means multiplying a by itself n times. For example, 2^4 = 2 x 2 x 2 x 2 = 16. The number a is called the base and n is the exponent (or power).
    Can I use decimal exponents?
    Yes. A decimal exponent like 2^0.5 is the same as √2 = 1.4142. Similarly, 8^(1/3) = ∛8 = 2. Fractional exponents connect powers and roots: a^(1/n) = n√a. This is one of the most useful identities in mathematics.
    Can I calculate the square root of a negative number?
    Not with real numbers. The square root of a negative number is an imaginary number (e.g., √(-1) = i). This calculator works with real numbers only. However, cube roots of negative numbers work fine: ∛(-8) = -2, because (-2) x (-2) x (-2) = -8.
    What is a^0 and why does it equal 1?
    Any non-zero number raised to the power of 0 equals 1. The reasoning: a^n / a^n = a^(n-n) = a^0, and since anything divided by itself is 1, a^0 = 1. So 5^0 = 1, 100^0 = 1, (-3)^0 = 1. The case of 0^0 is conventionally defined as 1 in combinatorics and most practical contexts.
    What are negative exponents?
    A negative exponent means the reciprocal: a^(-n) = 1/a^n. For example, 2^(-3) = 1/2^3 = 1/8 = 0.125. This is used extensively in scientific notation: 10^(-6) = 0.000001 = one millionth (the micro prefix).
    Where are powers and roots used in real life?
    Compound interest (principal x (1+r)^n), physics (inverse square law for gravity and light), geometry (area = side^2, volume = side^3), computer science (binary: 2^10 = 1024 bytes = 1 KB, 2^32 = 4.29 billion), statistics (standard deviation uses square roots), signal processing (decibels use log base 10), and engineering (stress calculations).
    What is the difference between 2^10 and 10^2?
    2^10 = 1024 (2 multiplied by itself 10 times). 10^2 = 100 (10 multiplied by itself 2 times). The base and exponent are not interchangeable - a^b is generally not equal to b^a (except for special cases like 2^4 = 4^2 = 16).
    How do powers relate to logarithms?
    Logarithms are the inverse of exponentiation. If 2^10 = 1024, then log2(1024) = 10. In other words: "2 raised to what power gives 1024? Answer: 10." Powers and logarithms are two sides of the same coin, useful for solving exponential equations.

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