This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
QUESTION 4. Write the following as simply as possible without using negative indices and as a single fraction. c) 3^-2x^-2(xy)^(1)/(2) + (1)/(x) Step 1: Convert terms with negative exponents to positive exponents. 3^-2 = (1)/(3^2) = (1)/(9) x^-2 = (1)/(x^2) The numerator of the first term becomes (1)/(9x^2). Step 2: Apply the exponent rule (ab)^n = a^n b^n to the denominator of the first term. (xy)^(1)/(2) = x^(1)/(2)y^(1)/(2) Step 3: Rewrite the first fraction by combining the results from Step 1 and Step 2. (1)/(9x^2)x^(1)/(2)y^(1)/(2) = (1)/(9x^2 x^1)2y^(1)/(2) Step 4: Combine the x terms in the denominator of the first fraction using the rule a^m · a^n = a^m+n. x^2 · x^(1)/(2) = x^2 + (1)/(2) = x^(4)/(2) + (1)/(2) = x^(5)/(2) So the first fraction is (1)/(9x^5)2y^(1)/(2). Step 5: The expression is now (1)/(9x^5)2y^(1)/(2) + (1)/(x). To combine these into a single fraction, find the least common denominator (LCD). The LCD for 9x^(5)/(2)y^(1)/(2) and x is 9x^(5)/(2)y^(1)/(2). Step 6: Adjust the second term to have the LCD. To change the denominator x to 9x^(5)/(2)y^(1)/(2), we multiply the numerator and denominator by 9x^(3)/(2)y^(1)/(2). (1)/(x) = (1 · (9x^3)/(2)y^(1)/(2))x · (9x^(3)/(2)y^(1)/(2)) = (9x^3)/(2)y^(1)/(2)9x^1+(3)/(2)y^(1)/(2) = (9x^3)/(2)y^(1)/(2)9x^(5)/(2)y^(1)/(2) Step 7: Add the two fractions. (1)/(9x^5)2y^(1)/(2) + (9x^3)/(2)y^(1)/(2)9x^(5)/(2)y^(1)/(2) = (1 + 9x^3)/(2)y^(1)/(2)9x^(5)/(2)y^(1)/(2) The final answer is (1 + 9x^3)/(2)y^(1)/(2)9x^(5)/(2)y^(1)/(2). Last free one today — make it count tomorrow, or type /upgrade for unlimited.