Properties of Exponents IIIPower to a PowerZero Power

Exponential Notation• Exponential Notation is nothing more than a shorthand

notation. Instead of writing out a number or variable times itself many, many times, we use exponential notation.

• The exponent tells us how many times the base is multiplied by itself.

Exponential Notation• Refer to your class notes for examples given in class.

Products of Monomials• Remember, “monomial” = “one term”• Monomials are products of numbers and variables.• Refer to your class notes for examples and non-examples of

monomials.

Products of Monomials• This comes back to exponential notation.• Suppose we are multiplying the following monomials:• (3x4y5)(-5x3y6)• Well, according to exponential notation this is really:• (3xxxxyyyyy)(-5xxxyyyyyy)• All total there are 7 x’s and 11 y’s.

Products of Monomials• Remember, we are multiplying, so -3 x 5 = -15.• Therefore, the final answer is -15x7y11.• Bottom line is, when you multiply two monomials you add the

exponents!• The reason for this is because with exponential notation, we

just have to remind ourselves that x4 means there are 4 x’s in the product and x3 means there are 3 more x’s to join them, making 7 total.

Products of Monomials• Refer to your notes for examples from class.

Quotient of Monomials• Again, we can make use of exponential notation.• Bottom line here is, we end up subtracting the exponents and

leave the variable wherever the exponent was greater.• Refer to your class notes for examples.

Negative Exponents• Negative exponents move the base from one part of the

fraction to the other. When this occurs, the exponent becomes positive.

• Remember to simplify the base before you move it. • Refer to your class notes for more examples.

Power to a Power• Here again, we are going to make use of exponential notation• Remember than x5 means that we are going to multiply x by

itself 5 times.• Well than (2x)5 means that we will be multiplying 2x by itself 5

times. Remember, the exponents reminds us how many times the base is being multiplied by itself.

• That means we will have:(2x)(2x)(2x)(2x)(2x) = 32x5

• Likewise, (3x2y3)3 means we will multiply 3x2y3 by itself 3 times. Every factor will have a 3, 2 x’s, and 3 y’s. So:

(3x2y3)(3x2y3)(3x2y3) = 27x6y9

Power to a Power• Point is, when you raise a monomial to a power, you multiply

the exponents of the base times the power of the monomial.

• See your notes for examples from class.

Zero Power• Any base raised to the power of 0 is equal to 1.

• a0=1

Properties of Exponents• a x a x a x … x a = an

• a-n = 1/an

• am x an = am+n

• am / an = am-n

• (am)n = amn

• a0 = 1• (ab)n = anbn

• (a/b)n = an/bn

• These same properties can be found onPages: 141, 152, 155, 331-332