Algebra Problems of the Day (Integrated Algebra Regents, January 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones. The Integrated Algebra Regents covered most of the same material as the current Algebra Regents, with a few differences.

More Regents problems.

Integrated Algebra Regents, January 2011

Part I: Each correct answer will receive 2 credits.


21. What is 3 √2 + √8 expressed in simplest radical form?

1) 3 √10
2) 3 √16
3) 5 √2
4) 7 √2

Answer: 3) 5 √2

Hopefully, you eliminated Choice (1) immediately, because radical addition doesn't work like that any more than 3² + 4² would equal 7².

Secondly, you should realize that √16 couldn't possible be simplest radical form because √16 is equal to 4, which is much simpler!

Like with fractions, you can only add radicals if they have the same radicand (number under the radical sign). Luckily, we can change the 8 into a 2.

√8 is not in simplest form because it factors in √((4)(2)), which is √4 √2, which is 2 √2.

So the sum of 3 √2 + √8 = 3 √2 + 2 √2 = 5 √2, which is Choice (3).

22. What is the slope of the line whose equation is 3x − 7y = 9?

1) -3/7
2) 3/7
3) -7/3
4) 7/3

Answer: 2) 3/7

The equation is in Standard Form, Ax + By = C, and the slope of a line in Standard Form is -A/B. So the slope is -(3)/(-7) = 3/7, which is Choice (2).

If you didn't remember that rule, you could have rewritten the equaion in slope-intercept form.

3x − 7y = 9

−7y = -3x + 9

y = (-3)/(-7)x + 9/(-7)

The slope is 3/7.

23. The figure shown below is composed of two rectangles and a quarter circle.

What is the area of this figure, to the nearest square centimeter?

1) 33
2) 37
3) 44
4) 58

Answer: 2) 37

Calculate the three pieces separately and then add them.

The rectangles are both 5 x 3, so each one is 15, and 15 + 15 = 30.

The quarter circle has a radius of 3, so A = 1/4 (3.14)(3)² = 7.065.

Add 30 and 7.065 and you get approximately 37.

24. The expression (10w³)² / (5w) is equivalent to

1) 2w⁵
2) 2w⁸
3) 20w⁵
4) 20w⁸

Answer: 3) 20w⁵

The exponent of 2 applies to everything in the parentheses. So (10w³)² = 10w³ * 10w³ = 100w⁶.

Divide 100w⁶ by 5w¹ and the quotient is 20w⁵, which is Choice (3).

You would have gotten a coefficient of 2 if you forgot to square the 10. You would have ended up with an exponent of 8 if you sqaured the exponent of 3 instead of doubling it before subtracting 1.

25. If ey/n + k = t, what is y in terms of e, n, k, and t?

1) y = (tn + k)/e
2) y = (tn - k)/e
3) y = n(t + k)/e
4) y = n(t - k)/e

Answer: 4) y = n(t - k)/e

use inverse operations to isolate y.

ey/n + k = t

ey/n = t - k

ey = n(t - k)

y = n(t - k)/e

The answer is Choice (4).

More to come. Comments and questions welcome.

More Regents problems.

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