A quotient of powers calls for the quotient rule: subtract the denominator’s exponent from the numerator’s for each matching base. You can check the number in front in Desmos by setting both variables to , making every variable power equal . Subtract the whole bottom exponent, even when it’s negative. Then rewrite any negative exponent as a reciprocal.
Hints
- Hint 1
For powers of , the quotient rule gives . The denominator’s exponent is negative here. What happens when you subtract a negative number?
- Hint 2
Use the same rule for the powers. The numerator’s exponent is already negative, so keep its sign when you subtract the denominator’s exponent. Will the resulting power of have a positive or negative exponent?
- Hint 3
Set both variables to in Desmos to isolate the coefficient, the number multiplying the powers. Once you have it, how can you write a negative exponent using only positive exponents?
Step-by-step
Subtract exponents for each base
Step 1Find the power of
The quotient rule says to subtract the bottom exponent from the top exponent for the same base. For , keep the negative bottom exponent in parentheses:
Subtract the negative number:
- Step 2
Find the power of
Apply the same rule to , including the minus sign on its top exponent:
Subtract :
- Step 3
Check the number in front
Type , , and the original fraction with in Desmos. Every power of is , so Desmos prints . That isolates the coefficient, . One input checks the coefficient, but it can’t prove two expressions match for every positive and .
- Step 4
Put the factors together
Use the coefficient and the two powers you found. Each base keeps its own exponent:
- Step 5
Write the answer with positive exponents
A negative exponent means a reciprocal, not a negative value:
Move that power below the fraction bar:
Since , the denominator isn’t zero. This is the equivalent expression. Choice B.