![]() Expressions with negative exponents in the numerator can be rewritten as expressions with positive exponents in the denominator: x − n = 1 x n.Any nonzero quantity raised to the 0 power is defined to be equal to 1: x 0 = 1.When a grouped quantity involving multiplication and division is raised to a power, apply that power to all of the factors in the numerator and the denominator: ( x y ) n = x n y n and ( x y ) n = x n y n.When raising powers to powers, multiply exponents: ( x m ) n = x m ⋅ n.When dividing two quantities with the same base, subtract exponents: x m x n = x m − n.When multiplying two quantities with the same base, add exponents: x m ⋅ x n = x m + n.In the following examples assume all variables are nonzero. In other words, any nonzero base raised to the zero power is defined to be equal to one. If the base is negative, then the result is still positive one. It is important to note that 0 0 is indeterminate. This leads us to the definition of zero as an exponent x 0 = 1 any nonzero base raised to the 0 power is defined to be 1., In general, given any nonzero real number x and integer n, Twenty-five divided by twenty-five is clearly equal to one, and when the quotient rule for exponents is applied, we see that a zero exponent results. Using the quotient rule for exponents, we can define what it means to have zero as an exponent. ( − 4 a 2 b c 4 ) 3 = ( − 4 a 2 b ) 3 ( c 4 ) 3 P o w e r r u l e f o r a q u o t i e n t = ( − 4 ) 3 ( a 2 ) 3 ( b ) 3 ( c 4 ) 3 P o w e r r u l e f o r a p r o d u c t = − 64 a 6 b 3 c 12 Given any positive integers m and n where x, y ≠ 0 we haveįirst apply the power rule for a quotient and then the power rule for a product. ![]() In summary, the rules of exponents streamline the process of working with algebraic expressions and will be used extensively as we move through our study of algebra. In general, this describes the use of the power rule for a product as well as the power rule for exponents. This is equivalent to raising each of the original grouped factors to the fourth power and applying the power rule. ( x 2 y 3 ) 4 = x 2 y 3 ⋅ x 2 y 3 ⋅ x 2 y 3 ⋅ x 2 y 3 = x 2 ⋅ x 2 ⋅ x 2 ⋅ x 2 ⋅ y 3 ⋅ y 3 ⋅ y 3 ⋅ y 3 C o m m u t a t i v e p r o p e r t y = x 2 + 2 + 2 + 2 ⋅ y 3 + 3 + 3 + 3 = x 8 y 12Īfter expanding, we are left with four factors of the product x 2 y 3. Now we consider raising grouped products to a power. This describes the power rule for exponents ( x m ) n = x m n a power raised to a power can be simplified by multiplying the exponents. ( x 6 ) 4 = x 6 ⋅ 4 = x 24 P o w e r r u l e f o r e x p o n e n t s ![]() Here we have 4 factors of x 6, which is equivalent to multiplying the exponents.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |