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44 results for “power”

Formulae (42)

Square of a Number

n^2 = n\times n

A number multiplied by itself.

Arithmetic & BasicsBasic

Cube of a Number

n^3 = n\times n\times n

A number multiplied by itself three times.

AlgebraIntermediate

Binomial Theorem

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expands any power of a binomial into a sum of terms.

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.

AlgebraIntermediate

Cube of a Binomial

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Expansion of a binomial raised to the third power.

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

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

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

Powers of complex numbers in trig form.

CalculusIntermediate

Power Rule (Derivative)

\frac{d}{dx}x^n = n x^{n-1}

Derivative of a power function.

CalculusIntermediate

Power Rule (Integral)

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \; n \ne -1

Antiderivative of a power function.

CalculusBasic

Power Rule

\frac{d}{dx}x^n = nx^{n-1}

Derivative of a power function.

CalculusBasic

Power Rule for Integration

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Antiderivative of a power function (n≠-1).

PhysicsBasic

Power

P = \frac{W}{t}

Rate of doing work.

PhysicsBasic

Electrical Power

P = VI = I^2R

Power dissipated in an electrical component.

PhysicsBasic

Power from Work

P = \frac{W}{t}

Rate of doing work.

Physics Laws & TheoremsAdvanced

Stefan–Boltzmann Law

j = \sigma T^4

Total energy radiated by a black body scales with the fourth power of temperature.

Optics & WavesBasic

Lens Power

P = \frac{1}{f}

Optical power in dioptres (f in metres).

Electronics & CircuitsBasic

Electrical Power (VI)

P = VI

Power dissipated by a component.

Electronics & CircuitsBasic

Power (I²R)

P = I^2 R

Power lost as heat in a resistor.

Electronics & CircuitsBasic

Power (V²/R)

P = \frac{V^2}{R}

Power from voltage and resistance.

Electronics & CircuitsAdvanced

AC Real Power

P = VI\cos\phi

Real power with a phase angle between V and I.

Electronics & CircuitsIntermediate

Decibel (Power)

dB = 10\log_{10}\frac{P_2}{P_1}

Power ratio expressed in decibels.

Electronics & CircuitsAdvanced

Apparent Power

S = \sqrt{P^2 + Q^2}

Combination of real and reactive power.

Biology & Life SciencesAdvanced

Kleiber's Law

q_0 = 70\,M^{0.75}

Metabolic rate scales with body mass to the ¾ power.

EngineeringBasic

Electrical Power

P = VI

Power in an electrical circuit.

EngineeringBasic

Efficiency

\eta = \frac{P_{out}}{P_{in}} \times 100\%

Ratio of useful output to input power.

EngineeringAdvanced

Electrical Power (3-phase)

P = \sqrt{3}\,V I \cos\phi

Real power in a balanced three-phase system.

EngineeringIntermediate

Power from Torque

P = \tau\omega

Mechanical power from torque and angular speed.

EngineeringBasic

Efficiency

\eta = \frac{P_{out}}{P_{in}}\times 100

Ratio of useful output to input power.

EngineeringIntermediate

Power Factor

PF = \frac{P}{S}

Ratio of real power to apparent power.

Aviation & AeronauticsIntermediate

Rate of Climb

ROC = \frac{\text{excess power}}{W}

Vertical speed from excess engine power.

Currency & ForexAdvanced

Purchasing Power Parity

S = \dfrac{P_{domestic}}{P_{foreign}}

The exchange rate that would equalize the price of an identical basket of goods across countries.

Currency & ForexAdvanced

Real Exchange Rate

RER = S \times \dfrac{P_{foreign}}{P_{domestic}}

The nominal rate adjusted for price levels, measuring relative purchasing power.

Environmental ScienceAdvanced

Wind Turbine Power

P = \frac{1}{2}\rho A v^3 C_p

Power extracted from wind by a turbine.

Unit ConversionsBasic

Horsepower to Watts

W = hp \times 745.7

Convert horsepower to watts.

Number TheoryIntermediate

Mersenne Number

M_p = 2^p - 1

Numbers one less than a power of two.

Math Tricks (2)

Subtract from 1000 Instantly

Subtract each digit from 9 except the last, which you subtract from 10. Works for any power of 10.

View trick

Anurupyena (Proportion Multiplication)

When numbers are near a convenient working base that is a multiple of a power of 10, use a proportional base to simplify the multiplication.

View trick