Prove that if a/b = c/d = e/f then (a^3b + 2c^2e - 3ae^2f)/(b^4 + 2d^2f - 3bf^3) = (ace)/(bdf)

TL;DR

Prove the equality of ratios using algebraic manipulation from a 1960s book.

Transcript

hey we're going to do a proof we're told that if a over b is equal to c over d and that's equal to e over f shoot that this is true so why does it say shoe instead of show um i don't actually know i should have done some research on that before making this video but i'm just sitting here doing some math and i've got a book with me it's a really old... Read More

Key Insights

• 🥶 Referencing older algebraic books can provide interesting and challenging problems for mathematical enthusiasts.
• 😑 Expressing ratios as variables can aid in simplifying algebraic proofs.
• 😑 Careful manipulation and simplification of expressions are essential steps in proving mathematical statements.

Questions & Answers

Q: What is the significance of the book "Higher Algebra" by Holland Knight in the video?

The book "Higher Algebra" by Holland Knight serves as the reference for the presented algebraic problem, emphasizing its importance in providing challenging algebra exercises.

Q: How does the narrator approach proving the equality of ratios a/b, c/d, and e/f?

The narrator sets the ratios as variable k = a/b = c/d = e/f and then expresses a, c, and e in terms of k, leading to a detailed algebraic manipulation to prove the equality.

Q: Why does the narrator express the numerators a, c, and e as variables in the proof?

By expressing the numerators as variables, namely a, c, and e, the narrator simplifies the manipulation process to illustrate the equality of the given ratios algebraically.

Q: How does the narrator simplify the final expression to prove the equality of the ratios?

The narrator carefully simplifies the expressions involving a, c, e, and the denominators to show that these ratios are indeed equal to ace divided by bdf, completing the proof.

Summary & Key Takeaways

• The video discusses proving the equality of ratios a/b, c/d, and e/f by expressing the numerators as variables a, c, and e.

• The narrator carefully manipulates the expressions to show that the given ratios are indeed equal to ace divided by bdf.

• The proof involves writing down the expressions, simplifying them, and showing that they are equal to the expected result.