One of the trickiest concepts in algebra involves the manipulation of exponents, or powers. Many times, problems will require you to simplify variables with exponents, or you will have to simplify an equation with exponents to solve it. To work with exponents, you need to know the basic exponent laws.
Adding and Subtracting with Non-like Terms
When a problem gives you two terms, or chunks, that do not have the exact same variables, or letters, raised to the exact same exponents, you cannot combine them. For instance, (4x^2)(y^3) + (6x^4)(y^2) could not be simplified (combined) further because the Xs and the Ys have different powers in each term.
Adding Like Terms
If two terms have the same variables raised to the exact same exponents, add their coefficients and use the answer as the new coefficient for the combined term. The exponents remain the same. For instance, 3x^2 + 5x^2 would turn into 8x^2.
Subtracting Like Terms
If two terms have the same variables raised to the exact same exponents, subtract the second coefficient from the first and use the answer as the new coefficient for the combined term. The powers themselves do not change. For example, 5y^3 - 7y^3 would simplify to -2y^3.
When multiplying two terms (it does not matter if they are like terms), multiply the coefficients together to get the new coefficient. Then, one at a time, add the powers of each variable to make the new powers. If you multiplied (6x^3z^2)(2xz^4), you would end up with 12x^4z^6.
Raising to Multiple Powers
When a term that includes variables with exponents is raised to another power, raise the coefficient to that power and multiply each existing power with the second one to get the new one. For instance, (5x^6y^2)^2 would simplify to 25x^12y^4.
First Power Exponent Rule
Anything raised to the first power stays the same. For example, 7^1 would just be 7 and (x^2r^3)^1 would simplify to x^2r^3.
Exponents of Zero
Anything raised to the power of 0 becomes the number 1. It does matter how complicated or big the term is. For instance, (5x^6y^2z^3)^0 would simplify to 1.
Dividing (When the Larger Exponent is on Top)
To divide when you have the same variable in the numerator and denominator, and the larger exponent is on top, subtract the bottom exponent from the top one and make the answer the new exponent of the variable on top. Then, eliminate the bottom variable. Reduce any coefficients like a fraction. If you were to do (3x^6)/(6x^2), you would end up with (x^4)/2.
Dividing (When the Smaller Exponent is on Top)
To divide when you have the same variable in the numerator and denominator, and the larger exponent is on the bottom, subtract the top exponent from the bottom one to make the new one on the bottom. Then, erase the variable from the numerator and reduce any coefficients like a fraction. If there are no variables left on top, leave a 1. For instance, (5z^2)/(15z^7) would become 1/(3z^5).
To eliminate negative exponents, put the term under 1 and change the exponent so that it is positive. For example, x^-6 is the same thing as 1/x^6. Flip a fraction raised to a negative exponent in order to make it positive: (2/3)^-3 means (3/2)^3. When division is involved, move variables from the bottom to the top or vice versa to make their exponents positive.