If x is a real number such that x^{3} = 729, then x^{2}+√x =?

Solution
To calculate the value of x^{2} + √x, first solve x^{3} = 729 for x.
The solution is the cube root of 729, which is 9.
Substitute 9 into the original expression, arriving at 9^{2} + √9.
This expression simplifies to 81 + 3, or 84.
What two numbers should be placed in the blanks below so that the difference between the consecutive numbers is the same?
13, __, __, 34

Solution
These four numbers will form an arithmetic sequence, a sequence in which each pair of successive terms differs by the same number.
To find the difference, define d as that difference, 13 as the first term, and 34 as the fourth term.
By definition, the second term is 13 + d. The fourth term, 34, can also be written as (13 + d + d) + d.
Using that expression, obtain the equation 34 = 13 + d + d + d, or 34 = 13 + 3d.
After subtracting 13 from both sides, divide by 3, which results in 7 = d. The difference is 7.
Thus the second term is 13 + 7, or 20, and the third term is 20 + 7, or 27.
If 6x − 3 = −5x + 7, then x =?

Solution
To solve for x in the equation 6x − 3 = −5x + 7, add 5x and 3 to both sides of the equation, which results in the equation 11x = 10.
Divide both sides by 11, which results in x = ^{10}⁄_{11}.
The expression a[(b − c) + d] is equivalent to:

Solution
To find an equivalent expression, simply distribute the a, as follows: ab − ac + ad. Remember to keep track of the negative sign.
A rectangular lot that measures 125 feet by 185 feet is completely fenced. What is the length, in feet, of the fence?

Solution
To find the length of fence needed to surround a rectangular lot 125 feet by 185 feet, calculate the perimeter.
The formula for perimeter of a rectangle is 2 times the sum of the length and width,
or P = 2(l + w).
Calculate the perimeter as follows:
2(125 + 185) = 2(310), or 620.
The oxygen saturation of a lake is found by dividing the amount of dissolved oxygen the lake water currently has per liter by the dissolved oxygen capacity per liter of the water, and then converting that number into a percent. If the lake currently has 6.4 milligrams of dissolved oxygen per liter of water and the dissolved oxygen capacity is 9.5 milligrams per liter, what is the oxygen saturation level of the lake, to the nearest percent?

Solution
To find the oxygen saturation level, divide the current number of milligrams per liter by the capacity milligrams per liter:\(\frac{6.4}{9.5}\).
Convert the result (0.6737) into a percent by multiplying by 100: 67.37% is approximately equal to 67%.
A student has earned the following scores on four 100point tests this marking period: 63, 72, 88, and 91. What score must the student earn on the fifth and final 100point test of the marking period to earn an average test grade of 80 for the five tests?

Solution
To find the score on the fifth 100point test that will yield an average score of 80, first calculate the total of the four scores already obtained: 63 + 72 + 88 + 91 = 314.
To obtain an average of 80 on 5 tests, the total score of all 5 tests must be 80×5, or 400.
The score needed on the last test is equivalent to 400 − 314, or 86.
Answer choice A is the average of the 4 scores, rounded to the nearest whole point.
Mr. Wilk is a high school math teacher whose salary is $33,660 for this school year, which has 180 days. In Mr. Wilk’s school district, substitute teachers are paid $85 per day. If Mr. Wilk takes a day off without pay and a substitute teacher is paid to teach his classes, how much less does the school district pay in salary by paying a substitute teacher instead of Mr. Wilk for that day?

Solution
To find Mr. Wilk’s pay per day, divide his annual salary, $33,660, by the total number of days he works, 180.
His pay per day is \(\frac{33,660}{180}\), or $187.
When Mr. Wilk takes a day off without pay and the school pays a substitute $85, the school district saves the difference in these amounts, 187−85, or $102.
Answer choice E, the most common incorrect answer, is simply Mr. Wilk’s pay per day and not the difference between his pay and a substitute’s pay.
4x^{3}× 3xy^{2}× 2xy^{2} is equivalent to:

Solution
To find an equivalent expression, multiply the constants (4 × 3 × 2 = 24),
combine the x terms (x^{3} × x × x) → x^{3+1+1} → x^{5}, and combine the y terms
(y^{2} × y^{2} → y^{2+2} → y^{4}).The result is 24x^{5}y^{4}.
The most common incorrect answers are F and H, which come from multiplying the exponents of the x and y terms instead of adding them.
If you chose G, you probably added the constants instead of multiplying them.
Shannon walked 1^{2}⁄_{3} miles on Wednesday and 2^{3}⁄_{5} miles on Thursday. What was the total distance, in miles, Shannon walked during those 2 days?

Solution
To find the total distance in miles that Shannon walked, add 1^{2}⁄_{3} and 2^{3}⁄_{5}.
To add mixed numbers, find the least common denominator.
The least common denominator of 3 and 5 is 3 × 5, or 15.
To convert ^{2}⁄_{3}, multiply by ^{5}⁄_{5}(hint:^{5}⁄_{5} = 1, and multiplication by 1 does not change the value of a number).
The result is ^{10}⁄_{15}. To convert ^{3}⁄_{5},multiply by ^{3}⁄_{3}. The result is 9^{9}⁄_{15}. To add 1^{10}⁄_{15} and 2^{9}⁄_{15}, first add 1 and 2 and then ^{10}⁄_{15} and ^{9}⁄_{15}. The result is 3^{19}⁄_{15}, which reduces to 4^{4}⁄_{15}.
Answer choice A is the most popular incorrect answer and comes from adding the whole numbers and then adding the numerators and the denominators separately.