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25 results for “exponent”

Formulae (25)

Order of Operations (PEMDAS)

P \to E \to (M \div D) \to (A \div S)

Parentheses, Exponents, Multiply/Divide, Add/Subtract.

AlgebraIntermediate

Logarithm Power Rule

\log_b(x^n) = n \cdot \log_b(x)

The logarithm of a power equals the exponent times the logarithm.

AlgebraBasic

Product of Powers

a^m \cdot a^n = a^{m+n}

Multiplying powers with the same base adds the exponents.

AlgebraBasic

Power of a Power

(a^m)^n = a^{mn}

Raising a power to a power multiplies the exponents.

AlgebraBasic

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Dividing powers with the same base subtracts exponents.

AlgebraBasic

Zero Exponent

a^0 = 1

Any nonzero number to the power zero equals one.

AlgebraBasic

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent means reciprocal.

AlgebraIntermediate

Fractional Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root.

AlgebraIntermediate

Exponential Growth / Decay

A = A_0 e^{kt}

Growth (k>0) or decay (k<0) over time.

AlgebraIntermediate

Logarithm Power Rule

\log_b(x^n) = n\log_b x

The log of a power brings the exponent to the front.

AlgebraIntermediate

Rational Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root and a power combined.

AlgebraBasic

Negative Exponent

a^{-n} = \frac{1}{a^n}

A negative exponent gives the reciprocal.

AlgebraBasic

Zero Exponent

a^0 = 1, \; a\neq 0

Any nonzero number to the power zero is one.

AlgebraBasic

Quotient of Powers

\frac{a^m}{a^n} = a^{m-n}

Divide powers with the same base by subtracting exponents.

TrigonometryAdvanced

Euler's Formula

e^{i\theta} = \cos\theta + i\sin\theta

Links complex exponentials to trigonometry.

CalculusIntermediate

Derivative of e^x

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusAdvanced

Maclaurin Series for e^x

e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}

Taylor series of the exponential at 0.

CalculusIntermediate

Derivative of Exponential

\frac{d}{dx}e^x = e^x

The exponential function is its own derivative.

CalculusIntermediate

Integral of Exponential

\int e^x\,dx = e^x + C

Antiderivative of the exponential function.

Biology & Life SciencesIntermediate

Exponential Population Growth

N_t = N_0 e^{rt}

Unlimited growth of a population over time.

Biology & Life SciencesIntermediate

Bacterial Doubling

N = N_0\times 2^{t/g}

Exponential bacterial growth by generations.

Environmental ScienceIntermediate

Exponential Population Growth

N = N_0 e^{rt}

Growth of a population at a constant rate.

Number TheoryAdvanced

Number of Divisors

d(n) = \prod (a_i + 1)

Count of divisors from prime factorization exponents.

Number TheoryAdvanced

Fermat Number

F_n = 2^{2^n} + 1

Numbers of a special exponential form.

Computer ScienceAdvanced

Exponential Time

O(2^n)

Runtime that doubles with each added element.