Question

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

Answer 1

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
\[2x^4-(-6x^4)=8x^4\]

Answer 2

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\[\frac{ 8x^5y^7 }{ 4x^3y^2 }=2x^{5-3}y^{7-2}=2x^2y^5\]

Answer 3

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.
\[(x^4+7)^0=1\]
\[5(x^6y^3)^0=5*1=5\]

Answer 4

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

Answer 5

Thanks! @peachpi