Mathematics Knowledge

Factor: 4y(3x + 2) − 2(3x + 2)

(4y − 2)(3x + 2) Any part of an expression that is common to every term in the expression can be factored out. In this case, (3x + 2) is a common term and can be factored out from both parts of the expression: 4y(3x + 2) − 2(3x + 2) = (3x + 2)[4y − 2] Notice that distributing (3x + 2) through [4y − 2] yields the original expression; this verifies the answer.

Solve: (−3)^3 =

-27 Remember the basic rules of exponents: n3 = n × n × n −33 = (−3) × (−3) × (−3) When you multiply a negative and a negative you get a positive: = 9 × (−3) When you multiply a negative and a positive you get a negative: = −27

6.6 * 10^-4

0.00066 Multiplying a decimal value by 10 raised to a power is equivalent to moving the decimal point to the left or right the number of times indicated by the power. In the case of a negative exponent, the decimal is moved to the left (this is the same as dividing by 10 a number of times). In the case of a positive exponent, the decimal is moved to the right (this is the same as multiplying by 10 a number of times). The negative exponent here, −4, indicates that the decimal point is to be moved to the left 4 places: 6.6 × 10−4 = 0.66 × 10−3 = 0.066 × 10−2 = 0.0066 × 10−1 = 0.00066 × 100 = 0.00066

What is the value of this numerical expression? √6^2 + 8^2

10 According to the order of operations (PEMDAS; parentheses, exponents, multiplication, division, addition, subtraction), the exponents within the square root must be evaluated first, then the sum can be found, and finally the square root can be evaluated. The expression simplifies to: √36+64= √=100 =10

What is the value of expression given below? 26-7(3+5)/4+2

14 We can use PEMDAS to evaluate the expression. (PEMDAS is a technique for remembering the order of operations. It stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) According to the rules, we must evaluate the expression in the parentheses first: 26 − 7(3 + 5) ÷ 4 + 2 = 26 − 7(8) ÷ 4 + 2 Then we perform multiplication and division in order from left to right as follows: 26 − 7(8) ÷ 4 + 2 = 26 − 56 ÷ 4 + 2 = 26 − 14 + 2 Finally, we perform addition and subtraction in order from left to right as follows: 26 − 14 + 2 = 12 + 2 = 14

If y= 3ab + 2b^3, what is y when a =1 and b =2?

22 Substitute the given values for a and b into the equation and then evaluate to find y: y = 3ab + 2b3 y = 3(1)(2) + 2(23) y = 6 + 2(8) y = 6 + 16 y = 22

Solve for x 3(x+1)=5(x-2)+7

3 Begin by distributing the 3 and the 5 through their respective parentheses, then combine like terms on each side of the equal sign: 3(x + 1) = 5(x − 2) + 7 3x + 3 = 5x − 10 + 7 3x + 3 = 5x − 3 Add 3 to both sides to maintain the equality: 3x + 6 = 5x Subtract 3x from both sides and then divide the resulting equation by 2 to solve for x as follows: 6 = 2x 3 = x

A rectangle is cut in half to create two square that each has an area of 25. What is the perimeter of the original rectangle?

30 The formula to find the area of a square is: Side* Side. If each square has an area of 25, then each side of the square must be 5 (because 5*5=25). So the the dimensions of the rectangle are 5*10. The perimeter is the sum of all sides, and is therefore 5+10+5+10=30

44 is 80% of what number?

55 Translate the question into an equation, using x to represent the unknown number: 44 = 0.8x Divide both sides by the decimal: 44 ÷ 0.8 = x x = 55

(7y^2 + 3xy - 9) - (2y^2 + 3xy -5)

5y^2 -4 When subtracting polynomials the first step is to distribute the negative sign through the parentheses. This changes the sign on each term inside the parentheses: 7y^2 + 3xy − 9 − 2y^2 − 3xy + 5 Arrange like terms next to each other (optional): 7y^2 − 2y^2+ 3xy − 3xy − 9 + 5 Combine like terms to compute the answer: 5y^2 − 4

Simplify the factorial: 6! =

720 To solve a factorial, recall that: n! equals the product of all the whole numbers from 1 to n, so n! = 1×2×3×...×n. 6! = 1×2×3×4×5×6 6! = 720

List A consists of the numbers {2, 9, 5, 1, 13}, and list B consists of the numbers {7, 4, 12, 15, 18}. If the two lists are combined, what is the median of the combined list?

8 Recall that the median of a set of data is the value located in the middle of the data set. In the case of a data set that contains an even number of numbers, the median is the average of the two middle numbers. Combine the 2 sets provided, and organize them in ascending order: {1, 2, 4, 5, 7, 9, 12, 13, 15, 18} Since there are an even number of items in the resulting list, the median is the average of the two middle numbers. Median = (7 + 9) ÷ 2 = 8

If 6w + 4 = 8w, then 4w =

8 The first step is to solve for w: 6w + 4 = 8w Subtract 6w from both sides: 4 = 2w Divide both sides by 2: 2 = w Now plug in the value of w: 4w = 4(2) = 8

In the figure below lines d and f are parallel and the angle measure are as given. What is the value of x?

85 Vertical angles are the angles opposite of each other that are created when two lines cross. Vertical angles are always equal. So, the angle vertical to the 35-degree angle also has a measure of 35 degrees. The supplement of the 120-degree angle has a measure of 60 degrees (all straight lines measure 180 degrees), so we then have a triangle with angles measuring 35, 60, and x degrees, as shown in the figure provided. Since the angles of a triangle add to 180 degrees: 35 + 60 + x = 180 x = 180 − 35 − 60 x = 85 degrees

Which of the following number is the greatest? 2/3 0.6 13/22 0.08

To determine which value is the greatest, convert the fractions to decimals and compare each value. The fraction 2/3 is equivalent to 2 ÷ 3 or 0.66; the fraction 13/22 evaluates to 0.59.

(x^6)(x^5)

x^11 Recall that the multiplication of terms sharing the same base can be simplified by adding the exponents: x^a * x^b = x(a + b). If this rule is forgotten, you can quickly derive it by considering the simple case of 21 * 22, which can be rewritten as 2 * 2 * 2, or 23. The expression in this case, then, can be simplified as: (x^6) (x^5) = x ^5+6 = x^11