Algebra 2 Problems of the Day

Algebra 2 is not my usual subject, but I do get asked about the problems occasionally. So I've decided to run a couple of Regents problems daily for a while. If there's a positive reaction (or at least, a lack of negative reaction), I may continue it.

More Algebra 2 problems.

January 2018
11. If n = sqrt(a⁵) and m = a, where a > 0, an expression n/m could be

(1) a^5/2
(2) a⁴
(3) (a²)^1/3
(4) (a³)^1/2
See image below

Answer: (4) (a³)^1/2
The square root of a value is the same as raising it to a power of 1/2, so n can be expressed as a^5/2.
Also, m can be expressed as a¹.
This means that n/m is the same as (a^5/2)/a¹.
When dividing, keep the base, subtract the exponents: a^{(5/2 - 1)} = a^(3/2)
A fractional exponent of 3/2 would mean take the square root of the third power, which is choice (4).



12. The solutions to x + 3 - (4 / (x - 1) ) = 5 are

Answer: (1) 3/2 + sqrt(17)/2
Follow the logic in the image:

Subtract 5 from each side, then add (4 / (x - 1)) to each side.
This will set up a rational equation. Multiply both sides by (x - 1) -- or "cross-multiply", if you prefer.
Multiply the binomials, then set up the quadratic equation to solve.
Use the Quadratic Formula to find the roots. Note that you have a positive discriminant, so the roots are real, and there is no "i" in the answer.
If you split the file fraction, you get choice (1).



Comments and questions welcome.