An equivalent expression gives the same value for every input. For multiple-choice expressions in one variable, test a convenient input in Desmos: a different output rules a choice out. Avoid inputs such as or when they make several powers look alike. A matching output alone doesn't prove equivalence, so check that only one choice remains.
Hints
- Hint 1
Choose an input that separates the choices. At , both and equal , so some choices tie. What happens if you use instead?
- Hint 2
An equivalent expression must match the original at . Enter that value and the original in Desmos, then compare its output with each choice's output. Any choice with a different value is out.
Step-by-step
Approach 1: Test a useful value in Desmos
- Step 1
Find the original expression's value
Use because it makes and different, unlike . Type , then type in Desmos. It shows , so an equivalent choice must also give at this input.
- Step 2
Compare all four outputs
Type the four choices on new lines, keeping . Desmos shows , , , and , in that order. Only one output matches the original's . One mismatch disproves equivalence; one match by itself doesn't prove it. Here the other three choices are ruled out, so the equivalent expression is . Choice B.
Approach 2: See why the terms cancel
- Step 1
Multiply by both parts of the first factor
The distributive property says each term in multiplies the entire second factor. Multiply by both and :
- Step 2
Pair terms with the same power
Terms with the same power of are like terms. Put each pair together to see their opposite signs:
- Step 3
Cancel the opposite pairs
Each pair adds to . The and the positive last term have no partners to cancel. Cancel the pairs:
So the product is equivalent to for every . Choice B.