Exponent Calculator

About the Exponent Calculator

Work out any power bˣ — with a whole-number, negative, fractional or decimal exponent — exactly wherever that is possible: 2^1000 with all 302 digits, 2^(−3) = 1/8, 8^(2/3) = 4, and 12^(1/2) = 2√3 in simplest radical form, together with the decimal value to as many as 100 significant digits. Every answer names the exponent law behind it: the zero and negative exponent rules, rational exponents as roots, the power of a power rule when the base is itself a power, and what a negative base does.

Solve bˣ = y finds the exponent, the base or the result: 8ˣ = 4 gives x = 2/3 exactly, 2ˣ = 10 gives x = log 10 ÷ log 2 ≈ 3.3219, and b² = 49 gives b = ±7. Exponent laws simplifies a product or quotient of powers such as x³ · x⁵, (2x³)² or (a/b)³ one law at a time.

Common uses

• Checking homework on indices, exponent rules and rational exponents.
• Getting the exact value of a large power, or a root in simplest radical form.
• Solving growth and decay equations such as 1.05ˣ = 2 (x ≈ 14.2).

The exponent laws

• Product rule: aᵐ · aⁿ = aᵐ⁺ⁿ, so x³ · x⁵ = x⁸.
• Quotient rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, so a⁷ ÷ a² = a⁵.
• Power of a power: (aᵐ)ⁿ = aᵐⁿ, so (y²)⁴ = y⁸.
• Power of a product: (ab)ⁿ = aⁿbⁿ, so (2x³)² = 4x⁶.
• Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ.
• Zero exponent: a⁰ = 1 for any a ≠ 0.
• Negative exponent: a⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/8.
• Rational exponent: a^(p/q) = (ᵠ√a)ᵖ = ᵠ√(aᵖ), so 8^(2/3) = (∛8)² = 4.

Fractional and decimal exponents

A decimal exponent is a fraction: 0.75 = 3/4, so 16^0.75 = (⁴√16)³ = 2³ = 8. When the root does not come out exactly, the answer is written in simplest radical form: every complete group of q equal prime factors comes out of the root, and a root in the denominator is cleared. So 2^(3/2) = 2√2 and (1/2)^(1/2) = √2/2. The decimal value is given as well.

Negative bases

(−2)³ = −8 and (−2)⁴ = 16: a negative base to an odd power is negative, to an even power positive. Without brackets, −2⁴ means −(2⁴) = −16. Odd roots of negative numbers are real — (−8)^(1/3) = −2 — but even roots are not: (−4)^(1/2) has no real value, and its principal complex value is 2i.

Solving bˣ = y

For the exponent, write both sides as powers of the same number when you can: 8 = 2³ and 4 = 2², so 2³ˣ = 2² and x = 2/3. Otherwise take logarithms of both sides: x = log y ÷ log b, with logarithms to any base, so 2ˣ = 10 gives x ≈ 3.3219. For the base, raise both sides to the power 1/x: b³ = 125 gives b = 125^(1/3) = 5. An even power has two real solutions: b² = 49 gives b = 7 or b = −7.

What is 0⁰?

0⁰ has no single agreed value. Algebra, combinatorics and most programming languages take it as 1 (the empty product), while in calculus 0⁰ is an indeterminate form. The calculator explains this rather than choosing for you.

Limitations

• Exact answers are written out up to 100,000 digits; bigger powers are given in scientific notation. For exact arithmetic on even larger numbers, use the big number calculator.
• Simplest radical form needs the prime factors of the base: a very large base that cannot be factorised quickly, or a root with an index above 1,000 (an exponent such as 0.123457), gets a decimal value only.
• Exponent laws mode takes number exponents (not letters, as in aⁿ), single-letter variables, and products and quotients — not sums. A power applies only to the number right after ^, so x^2/3 means x² ÷ 3: write a fractional exponent in brackets, x^(2/3). With fractional exponents, letters are taken to be positive numbers, as the laws require.
• Decimal values are rounded to the significant digits you choose (up to 100), half to even.

Privacy

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