Unit 2 Laws of Exponents 8437 Exponent or Power Base

You can only add like terms! You learned months ago you can only add like terms. When there are like terms (as determined by the variables & exponents) you just add the coefficients. 2n3 + 6n4 6m3 + 9n3 12n3 -16n3=

You can only add like terms! You learned months ago you can only add like terms. When there are like terms (as determined by the variables & exponents) you just add the coefficients. 2n3 + 6n4 This is already simplified. They are not like terms. You cannot add them. 6m3 + 9n3 This is already simplified. They are not like terms. You cannot add them. 12n3 -16n3= -4n3 This can be simplified since they are like terms. You add the coefficients and re-write the variable part.

(5t)(-30t2) = (8xz)(-10y)(-2yz2) = You can multiply anything. You multiply the coefficients, write all of the variables and then add the exponents. (5t)(-30t2) = (8xz)(-10y)(-2yz2) =

(5t)(-30t2) = -150t3 (8xz)(-10y)(-2yz2) = 160xy2z3 You can multiply anything. You multiply the coefficients, write all of the variables and then add the exponents. (5t)(-30t2) = -150t3 (8xz)(-10y)(-2yz2) = 160xy2z3

PRODUCT OF POWERS For all real numbers b and all positive numbers m & n, bm * bn = bm+n This means to multiply powers with the same base, you add the exponents and keep the same base. a3(a4) 7x3  3x4

PRODUCT OF POWERS For all real numbers b and all positive numbers m & n, bm * bn = bm+n This means to multiply powers with the same base, you add the exponents and keep the same base. a3(a4) means (aaa)  (aaaa) = a7 7x3  3x4 means (7xxx)  (3xxxx) = 21x7

POWER OF A POWER PROPERTY POWER OF A POWER PROPERTY For all non-zero real numbers x and all integers m & n, (xm)n = xmn This means to raise a power to a power with the same base, you multiply the powers and leave the base alone. (a3)4 (5a3)2

POWER OF A POWER PROPERTY POWER OF A POWER PROPERTY For all non-zero real numbers x and all integers m & n, (xm)n = xmn This means to raise a power to a power with the same base, you multiply the powers and leave the base alone. (a3)4 means (a3) (a3) (a3) (a3) = (aaa) (aaa) (aaa) (aaa) = a12 (5a3)2 means (5a3) (5a3) = (5aaa)  (5aaa) =25 a6

POWER OF A PRODUCT PROPERTY For all nonzero real numbers x & y, and all integers m &n (xy)n =xnyn This means you raise everything that is on the inside of the parentheses to the power of the exponent. (3b2)3 4(x)2 5(y3)4

POWER OF A PRODUCT PROPERTY For all nonzero real numbers x & y, and all integers m &n (xy)n =xnyn This means you raise everything that is on the inside of the parentheses to the power of the exponent. (3b2)3 means (3b2) (3b2) (3b2) = (3bb) (3bb) (3bb)= 27b6 Since the 3 was inside the parentheses it also was raised to that power. 4(x)2 means 4xx= 4x2 Since the 4 was outside of the parentheses it was not raised to that power. 5(y3)4 means 5 (y3) (y3) (y3) (y3) = 5y12 Since the 5 was outside of the parentheses it was not raised to the 4th power.

POWERS OF –1 Even powers of –1 are equal to 1 Odd Powers of –1 are equal to –1 (-y)4 -y100 (-3y)3

POWERS OF –1 Even powers of –1 are equal to 1 Odd Powers of –1 are equal to –1 (-y)4 means (-1*y) (-1*y) (-1*y) (-1*y) = y4 Since that negative was inside ( ) the resulting product is positive. -y100 is already simplified. The negative is not inside ( ) so you do not raise it to the power. It stays negative. (-3y)3 means (-3y) (-3y) (-3y) = -27y3

QUOTIENT OF POWERS PROPERTY For all nonzero numbers x and all integers m & n , xm = xm-n xn This means, if you are dividing the same base, you actually subtract the exponents.

QUOTIENT OF POWERS PROPERTY For all nonzero numbers x and all integers m & n , xm = xm-n xn This means, if you are dividing the same base, you actually subtract the exponents.

If you subtract and get a negative number for the exponent, you will write the term as the denominator of a fraction with a numerator of 1. z2 z10 zz zzzzzzzzzz 1 zzzzzzzz z8 z -8 =

If you subtract and get a negative number for the exponent, you will write the term as the denominator of a fraction with a numerator of 1. z2 z10 zz zzzzzzzzzz 1 zzzzzzzz z8 z -8 = DEFINITION OF A NEGATIVE EXPONENT For all nonzero numbers x and all integers n , 1 xn x-n =

ZERO AS AN EXPONENT For any nonzero number x, x0 = 1 This means, any number raised to the zero power is equal to 1.

POWER OF A FRACTION PROPERTY For all real numbers a, b, n where n > 0 and b ≠0, a n = bn This means, if the exponents will be used on the numerator and on the denominator. x 3 x3 x3 = = 4 43 64

Some problems have several steps Some problems have several steps. Use your rules and multiply the exponents in that numerator and then subtract the exponents when you were dividing.

You can always write these problems out the long way if you get stuck. It is helpful to study & learn the rules so you can solve them faster. However, it is more important that you understand how we are getting the correct answers than just stating which exponent rule we used.