Question - Solving an Expression with Sum of Cubes
Solution:
The image is rotated; I'll provide the steps to reach the solution as if the image were in the correct orientation.Given: a^2/b + b^2/a = ?To find the above expression, we combine the terms over a common denominator:Step 1: a^2/b + b^2/a = (a^3 + b^3) / (ab)Then, by applying the sum of cubes formula a^3 + b^3 = (a + b)(a^2 - ab + b^2), we simplify the expression:Step 2: (a^3 + b^3) / (ab) = [(a + b)(a^2 - ab + b^2)] / (ab)Finally, we factor out a from the first term and b from the second term in the numerator:Step 3: [(a + b)(a^2 - ab + b^2)] / (ab) = (a + b)(a - b + b) / b + (a + b)(a + a - b) / aThis simplifies to:Step 4: (a + b)(a) / b + (a + b)(b) / aNow we split the terms to simplify further:Step 5: a(a + b) / b + b(a + b) / aSeparate the terms:Step 6: a^2/b + ab/b + ab/a + b^2/aFurther simplification:Step 7: a^2/b + a + a + b^2/aCombine like terms:Step 8: a^2/b + 2a + b^2/aAnd if required to write as a single fraction:Step 9: (a^3 + 2a^2b + b^3) / abThis represents the expression in terms of a single fraction, but the final simplified form is as mentioned in step 8:Final Answer: a^2/b + 2a + b^2/a
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com