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Home › Math & Conversions › Core Math & Algebra › Exponent Exponent Calculator Compute \(a^b\) quickly and safely. Supports positive, zero, negative and fractional exponents, and shows results in standard and scientific notation with clear explanations. This page covers the mathematical meaning of exponents and exponentiation, not the interview platform or consulting firms that share the name “Exponent”. Core Math & Algebra Interactive exponentiation tool Enter a base \(a\) and an exponent \(b\) and the calculator will evaluate \(a^b\), check domain issues (like \(0^0\) or negative bases with non-integer exponents), and present the result in multiple formats. Base \(a\) Any real number. For negative bases, only integer exponents are allowed. Exponent \(b\) Accepts integers, decimals, and simple fractions like 1/2 or 3/4. Decimal places 4 decimals 8 decimals 12 decimals Only affects how the result is displayed, not the internal precision. Options Show repeated-multiplication steps for small integer exponents Always show scientific notation \(m \times 10^k\) Calculate \(a^b\) Clear Example: \(2^{10}\) Example: \(10^{-3}\) Example: \(9^{1/2}\) Results Main value Scientific notation Interpretation Step-by-step multiplication For larger exponents, repeated multiplication is not shown explicitly; the calculator still uses an efficient power algorithm internally. What is an exponent? In mathematics, an exponent tells you how many times to multiply a base by itself. The expression \(a^b\) is read “\(a\) to the power of \(b\)” or “\(a\) raised to \(b\)”. For example: \(2^3 = 2 \cdot 2 \cdot 2 = 8\) \(10^4 = 10\,000\) \(5^1 = 5\) Exponentiation extends this idea beyond simple repeated multiplication, allowing zero , negative , and even fractional exponents. Special exponent cases Zero and one \(a^1 = a\) for any real \(a\). \(a^0 = 1\) for any nonzero \(a\). This is consistent with the laws of exponents. \(0^0\) is indeterminate in many contexts and is left undefined in this calculator. Negative exponents A negative exponent means reciprocal . For a nonzero base \(a\) and positive integer \(n\): \(a^{-n} = \dfrac{1}{a^n}\). Examples: \(2^{-3} = 1/2^3 = 1/8 = 0.125\) \(10^{-2} = 1/100 = 0.01\) Fractional exponents Fractional exponents connect exponentiation with roots. For a positive base \(a\) and integers \(p\) and \(q>0\): \(a^{p/q} = \left(\sqrt[q]{a}\right)^p = \sqrt[q]{a^p}\). Examples: \(9^{1/2} = \sqrt{9} = 3\) \(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\) \(16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8\) Basic laws of exponents These rules hold for a wide range of exponents (with the usual domain caveats such as nonzero base when dividing): \(a^m \cdot a^n = a^{m+n}\) (same base, add exponents) \(\dfrac{a^m}{a^n} = a^{m-n}\) (subtract exponents) \((a^m)^n = a^{mn}\) (power of a power) \((ab)^n = a^n b^n\) (power of a product) \(\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}\) (power of a quotient) The calculator respects these relationships when simplifying and interpreting your input. Exponents and scientific notation Exponents are central to scientific notation , which expresses numbers as \(m \times 10^k\) with \(1 \le |m| < 10\) and integer \(k\). This makes it easy to work with very large or very small values. For example: \(3\,000 = 3 \times 10^3\) \(0.000045 = 4.5 \times 10^{-5}\) The exponent calculator automatically converts your result into this form so you can quickly see the order of magnitude. Exponent – FAQ Are all combinations of base and exponent valid over the real numbers? No. For example, \(0^0\) and \(0^{-1}\) are undefined, and negative bases with non-integer exponents typically produce complex numbers. This tool focuses on real-valued exponentiation and will warn you when an input falls outside that domain. Why is 0 raised to a positive exponent allowed? For any positive exponent \(b\), \(0^b = 0\) is consistent with the limit behavior and algebraic rules. Problems appear only with zero or negative exponents on a zero base, where division by zero or indeterminate forms arise. How does this relate to logarithms? Logarithms are the inverse of exponentiation. If \(a^b = x\) with \(a>0\) and \(a \ne 1\), then \(b = \log_a x\). While this page focuses on computing powers, the same concepts underpin log and exponential functions used in finance, statistics, and physics. Does the calculator use exact arithmetic? Internally, it uses standard floating-point arithmetic, which is approximate. For many everyday values and exponents this is more than sufficient. The decimal-places setting only controls how the result is displayed, not the underlying precision. Can I use this for teaching and exam work? Yes. The tool is designed as a teaching aid: it highlights domain issues, shows scientific notation, and can display repeated-multiplication steps for small integer exponents. For formal work, you may still want to show the algebra manually, but the calculator is excellent for checking results. Frequently Asked Questions How accurate is this exponent calculator? It uses the browser’s IEEE-754 floating-point arithmetic, which is the same standard used in most scientific software and programming languages. Results are usually accurate to many decimal places; any rounding errors are far smaller than typical measurement or modelling uncertainties. What should I watch out for when using very large exponents? Extremely large positive exponents can overflow to infinity, while very negative exponents can underflow to zero. When that happens, the calculator will warn you. In such regimes, using logarithms or symbolic manipulation is often more appropriate. Can this calculator handle complex exponents? No, this version is restricted to real-valued exponentiation. Complex exponents require careful branch choices and complex analysis results, which go beyond the scope of a general-purpose calculator aimed at everyday math, science, and engineering users. Related Core Math & Algebra tools Explore more algebra and function tools that naturally pair with exponents and powers. Power & root calculator Logarithm calculator Scientific notation Exponential growth rate Modular arithmetic Set theory Interval notation Continued fraction Convolution All Math & Conversions tools Checklist for safe exponent use Check whether the base is zero, negative, or positive. Identify if the exponent is integer, negative, or fractional. For negative bases, ensure the exponent is an integer. Avoid undefined forms like \(0^0\) or \(0^{-1}\). Use scientific notation when communicating very large or small results.
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