Question

Which expressions are equivalent?
mc015-1.jpg and mc015-2.jpg
mc015-3.jpg and mc015-4.jpg
mc015-5.jpg and mc015-6.jpg
mc015-7.jpg and mc015-8.jpg

Answer 1

I can't see the images you referenced. Please either

- paste each expression as text here (e.g. (x^2-1)/(x-1) ), or
- upload the image files, or
- type a brief description of each expression.

If you want to check equivalence yourself, here are quick methods you can use:

1. Algebraic simplification
- Expand or factor both expressions, then simplify (cancel common factors, combine like terms). If simplified forms match exactly and have the same domain, they are equivalent.
- Watch domain restrictions (division by zero, even roots of negative numbers, logs of nonpositive numbers). Two expressions can simplify to the same algebraic form but differ where they are defined.

2. Use identities (for trigonometric, logarithmic, exponential expressions)
- Apply standard identities (e.g. sin^2+cos^2=1, log rules, exponent rules) and simplify both sides.

3. Numerical testing (fast check)
- Evaluate both expressions at several random values of the variable(s) that lie in the intersection of their domains. If results match for many distinct values, they are likely equivalent; if a counterexample is found, they are not equivalent. This doesn’t prove equivalence symbolically but is a good quick test.

Example:
- To check (x^2−1)/(x−1) vs x+1:
- Factor numerator: (x−1)(x+1). Cancel (x−1) to get x+1, but note x≠1 (domain restriction). So they are algebraically equivalent except at x=1 (the first expression is undefined there).

Send me the actual expressions (or clear images) and I will determine which pairs are equivalent and point out any domain differences.