All exponent rules? Adding, subtracting, multiplying, dividing, etc...

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All exponent rules? Adding, subtracting, multiplying, dividing, etc...

Mathematics
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You can only add and subtract like terms, meaning each term has the same variables to the same power. For example you can add and subtract \(2x^4\) and \(-6x^4\) by adding or subtracting the coefficients and leaving the exponents the same. Addition \[2x^4+(-6x^4)=-4x^4\] Subtraction \[2x^4-(-6x^4)=8x^4\]
With multiplication and division, the exponents change. Multiplication (multiply coefficients, add exponents) \[x^mx^n=x^{m+n}\] Example \[(8x^5y^7)(4x^3y^2)=32x^{5+3}y^{7+2}=32x^8y^9\] Division (divide coefficients, subtract exponents) \[\frac{ x^m }{ x^n }=x^{m-n}\] Example\[\frac{ 8x^5y^7 }{ 4x^3y^2 }=2x^{5-3}y^{7-2}=2x^2y^5\]
Power to a power (multiply the exponents) \[(x^m)^n=x^{mn}\] Example \[(x^3y^5)^2=x^{3*2}y^{5*2}=x^6y^{10}\] Negative exponents (take the reciprocal, change the exponent to positive) \[x^{-m}=\left( \frac{ 1 }{ x } \right)^m\] Example \[(3x)^-5=\left( \frac{ 1 }{ 3x } \right)^5\] \[\left( \frac{ 1 }{ 2 } \right)^{-3}=2^3=8\] Anything to the 0 power is 1. \[(x^4+7)^0=1\] \[5(x^6y^3)^0=5*1=5\]

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Other answers:

Rational exponents / Radicals \[\sqrt[n]{x^m}=x^\frac{ m }{ n }\] Example \[\sqrt{3x}=(3x)^\frac{ 1 }{ 2 }\] \[\sqrt[4]{(12x)^5}=(12x)^\frac{ 5 }{ 4 }\]
Thanks! @peachpi

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