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Let a and b be real numbers and m and n be integers a. 13°+(1/3)^−2+(1/9)^−1 b. (14a²b^7/2a^5b)^−2 c. (−3x)^−4(4x^−2y^3)^3

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Final answer:

To simplify the expressions: a. 367/39, b. 2a^6/b^12 * 1/25, c. 64y^9/81x^10.

Step-by-step explanation:

a. To simplify the expression, we can convert the negative exponents to positive exponents by taking the reciprocal of each term. 13° can be written as 1/13, (1/3)^-2 can be written as (3/1)^2, and (1/9)^-1 can be written as (9/1)^1. Simplifying further, we get 1/13 + 9/1 + 1/3. Adding these fractions, we get 1/13 + 27/3 + 1/3 = 1/13 + 28/3 = (1/13 + 28/3)(3/3) = (3 + 364)/39 = 367/39.

b. To simplify the expression, we can combine the variables and exponents. (14a²b^7/2a^5b)^-2 = 14^-2 * (a²)^-2 * (b^7)^-2 / (2a^5)^-2 * b^-2. Simplifying further, we get 1/14^2 * 1/(a^2*2)^2 * 1/(b^7*2)^2 / 1/(2^2*a^5)^2 * 1/b^2 = 1/196 * 1/a^4 * 1/b^14 / 1/16 * 1/a^10 * 1/b^2 = 16/196 * a^10/a^4 * b^2/b^14 = 2/25 * a^6 * 1/b^12 = 2a^6/b^12 * 1/25.

c. To simplify the expression, we can apply the power of a power property and distribute the exponents. (-3x)^-4(4x^-2y^3)^3 = (-3)^-4 * x^-4 * (4)^3 * (x^-2)^3 * (y^3)^3 = 1/(-3)^4 * 1/x^4 * 4^3 * x^-6 * y^9 = 1/81 * 1/x^4 * 64 * 1/x^6 * y^9 = 64/81 * 1/(x^4 * x^6) * y^9 = 64/81 * 1/x^10 * y^9 = 64y^9 / 81x^10.

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