Exponents and Polynomials
Multiplication with Exponents
Definition of power: For n ∈ { natural numbers}, a > 0 and a ≠ 1,
a^{ n} = a Â· a Â· a Â· a Â·...Â· a (n factors of a)
[NOTE: We also say "a to the n^{th} power"
Rule of signs: Remember (1)(1) = +1 and (1)(1)(1) =  1
Generally, count the number of minus signs in a product. Attach "+" for even and
"â€“" for odd
NOTE: The exponent affects only the number that it immediately touches so you must always
put negative numbers in parentheses ( ).
Example 1:
1. 5^{3} = 5Â·5Â·5 = 125 base = 5, exponent = 3 Say
"five to the third power"
2. â€“2^{4} = â€“ 2Â·2Â·2Â·2 = â€“ 16 base = 2, exponent = 4
"negative of two to the fourth power"
3. ( 2)^{4} = (2)Â·(2)Â·(2)Â·(2) = 16 base = 2, exponent = 4
"negative two to the fourth power"
4.
exponent = 3
"negative twofifths to the third power"
Multiplication of "powers with the same base"
Example 2:
1. 2^{4} Â·2^{3} = ( 2Â·2Â·2Â·2 )
Ã—(2Â·2Â·2) = 2^{7} Notice that 4 + 3 = 7
2. ( 2)^{4}Â·( 2)^{3}= [(2)Â·(2)Â·(2)Â·(2)] Ã— [(2)Â·(2)Â·(2)] = [16] Ã—[â€“ 8] = â€“ 128 = ( 2)^{7}
1. For base a and m, n ∈ { natural numbers}
→ a^{m}
Â·a^{n} = a^{m + n} → "Multiplication is shortcut addition" Add exponents.
Example 3:
1. 2^{5} Â·2^{4} = 2^{9} = 512
2. x^{7} Â·x^{9} = x^{16}
Powering of "powers with the same base"
For base a and m, n ∈ { natural numbers}
→ (a^{m})^{n} = a^{m
Â· n} →
[Powering is shortcut multiplication â€“ multiply exponents.]
Say: "To raise a power to a power, ^{multiply exponents}."
Example 4:
1. (2^{4}
)^{3} = (2Â·2Â·2Â·2) Ã—(2Â·2Â·2Â·2) Ã—(2Â·2Â·2Â·2) = 2^{12} Notice that 4 Ã—3 = 12
2. (( 2)^{3})^{ 3} = [(2)Â·(2)Â·(2)] Ã—[(2)Â·(2)Â·(2)] Ã—[(2)Â·(2)Â·(2)]= (2)^{9} = 2^{9} =  512
[negative raised to an odd power]
Notice that 3 Ã—3 = 9
Powering of "products with different bases"
For bases a and b, and n ∈ { natural numbers}
→ (aÂ·b)^{n} = a^{n}
Â·b^{n} = → say â€œThe power of product is the product of the powers.â€
Example 5:
1. (2Â·3)^{3} = (2Â·3)Ã—(2Â·3)Ã—(2Â·3) = (2Â·2Â·2 ) Ã—(3Â·3Â·3) = (2^{3})(3^{3})
2. (2Â·x^{2}y^{3})^{4} = (2)^{4}Â·(x^{2})^{4}(y^{3})^{4} = 16Â·(x^{2Ã—4})(y^{3Ã—4}) = 16
Â·x^{8}
Â·y^{12
}
