Short Answer.Raise the quanity in parenthesis to the indicated exponents and simplify.Express answers with positive exponents.(-27x^0y^-2/54x^-5y^-4)^-2
To raise the quantity in parentheses to the indicated exponents and simplify, follow these steps:
1. Begin by simplifying the terms within the parentheses. Start by simplifying the numerical coefficients: -27/54 = -1/2.
The expression now becomes: (-1/2x^0y^-2/54x^-5y^-4)^-2.
2. Deal with the exponents. Remember that any number (except 0) raised to the power of 0 is equal to 1. Therefore, x^0 = 1.
The expression becomes: (-1/2y^-2/54x^-5y^-4)^-2.
3. To simplify further, use the rule that states when we have a negative exponent, it can be moved to the denominator as a positive exponent. So, move the negative exponents:
(-1/2y^-2/54x^-5y^-4)^-2
= (-1/2 * 54 * y^2 * x^5 * y^4)^-2.
4. Multiply the numerical coefficients together: -1/2 * 54 = -27.
The expression now becomes: (-27 * y^2 * x^5 * y^4)^-2.
5. Combine the variables: y^2 * y^4 = y^(2+4) = y^6.
The expression becomes: (-27 * y^6 * x^5)^-2.
6. Finally, apply the power rule for exponents. Raise each term inside the parentheses to the power of -2:
(-27 * y^6 * x^5)^-2 = (-27^-2 * (y^6)^-2 * (x^5)^-2).
7. Simplify each term inside the parentheses:
-27^-2 = (-1/27)^-2 = 1/(-1/27)^2 = 1/(-1^2/27^2) = 1/(1/729) = 729.
(y^6)^-2 = y^(6*-2) = y^-12 = 1/y^12.
(x^5)^-2 = x^(5*-2) = x^-10 = 1/x^10.
8. Replace the terms inside the parentheses with their simplifications:
(-27 * y^6 * x^5)^-2 = (729 * (1/y^12) * (1/x^10)).
9. Now, rewrite the expression:
(729 * (1/y^12) * (1/x^10)) = 729/(xy)^10 * 1/y^12.
This is the simplified expression after raising the quantity in parentheses to the indicated exponents.