# Solutions and Explanations to Intermediate Algebra Questions in Sample 1

Solutions and full explanations of intermediate algebra questions in sample 1 are presented.

 Write 230,000,000,000 in scientific notation. Solution Write the given number in the form a × 10 n , where a is a real number such that 1 ≤ |a| < 10 and n is an integer. 230,000,000,000 = 2.3 × 100,000,000,000 = 2.3 × 10 11 Evaluate: 30 - 12÷3×2 = Solution According to order of operations, 12÷3×2 (division and multiplication) is done first from left to right 12÷3×2 = 4 × 2 = 8 Hence 30 - 12÷3×2 = 30 - 8 = 22 Evaluate: |4 - 8(3 - 12)| - |5 - 11| = Solution According to order of operations, inner brackets first. Hence |4 - 8(3 - 12)| - |5 - 11| = |4 - 8*(-9)| - |5 - 11| According to order of operations, multiplication within absolute value signs (which may be considered as brackets when it comes to order of operations) next. Hence = |4 + 72| - |5 - 11| = |76| - |-6| = 76 - 6 = 70 Evaluate: -18 + 4(6 ÷ 2)2 Solution According to order of operations, inner brackets first. Hence -18 + 4(6 ÷ 2)2 = -18 + 4(3)2 According to order of operations, power next. Hence = -18 + 4*9 According to order of operations, multiplication next. Hence = -18 + 36 = 18 Evaluate: 11 + sqrt(- 4 + 6×4÷3) Solution According to order of operations, inner brackets first where 6×4÷3 is first calculated since it has a multiplication and a division. 6×4÷3 = 24÷3 = 8 Hence 11 + sqrt(- 4 + 6×4÷3) = 11 + sqrt(- 4 + 8) = 11 + sqrt(4) = 11 + 2 = 13 Simplify: 12x3 - 3(2x3 + 4x -1) - 5x + 7 Solution First expand the term - 3(2x3 + 4x -1) 12x3 - 3(2x3 + 4x -1) - 5x + 7 = 12x3 - 6 x3 - 12 x + 3 - 5x + 7 Group like terms = 6 x3 - 17 x + 10 Simplify:$(\dfrac{x^4}{x^3})^3$ Solution Use quotient of powers formula $\dfrac{x^m}{x^n}=x^{m-n}$ to simplify $\dfrac{x^4}{x^3}$. $(\dfrac{x^4}{x^3})^3=(x^{4-3})^3=x^3$ Simplify: $\dfrac{(3x^2y^{-2})^3}{(9xy^3)^3}$ Solution Use power of quotient formula $\dfrac{a^m}{b^m}=(\dfrac{a}{b})^m$ $\dfrac{(3x^2y^{-2})^3}{(9xy^3)^3}= (\dfrac{3 x^2y^{-2}}{9 x y^3})^3$ $= (\dfrac{x^{2-1}}{3y^{3+2}})^3$ $= (\dfrac{x}{3y^5})^3$ $= \dfrac{x^3}{27y^{15}}$ Simplify: $\dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3}$ Solution Note that the above expression is defined when neither $x$ nor $y$ is equal to zero and therefore $(x^3 + y)^0 = 1$. Hence $\dfrac{(2x^{-3}y^4)^3(x^3 + y)^0}{(4xy^{-2})^3} = \dfrac{(2x^{-3}y^4)^3}{(4x y^{-2})^3}$ $=(\dfrac{2x^{-3}y^4}{4x y^{-2}})^3$ $= ((1/2) \dfrac{y^{4+2}}{x^{1+3}})^3$ $= (1/8) (\dfrac{y^6}{x^4})^3$ $= (1/8) \dfrac{y^{18}}{x^{12}}$ Write as a mathematical inequality:"9 is less than the product of M and N". Solution 9 < M × N Find the slope of the line perpendicular to the line y = (1/3)x - 7 Solution Two lines are perpendicular if the product of their slopes is equal to -1. The slope of the given line is equal to 1 / 3. If m is the slope of the line perpendicular to the given line, then m × (1/3) = -1 Solve for m m = - 3 Write an equation of the line with slope -3 and y-intercept (0 , -5). Solution y = m x + b is the general form of the equation of a line in slope intercept form. Hence for m = -3 and b = -5, we have the equation y = - 3 x - 5 Solve the equation: -5 x + 20 = 25 Solution Subtract 20 to both sides -5 x + 20 - 20 = 25 - 20 Simplify - 5 x = 5 Divide both side s b y -5 x = - 1 Solve the inequality: -3x + 4 < -8 Solution Subtract 4 from both sides - 3 x + 4 - 4 < - 8 - 4 Simplify - 3 x < - 12 Divide both sides b y - 3 and change symbol of inequality x > 4 Solve the equation: 2x2 - 32 = 0 Solution Add 32 both sides 2x2 - 32 + 32 = 0 + 32 Simplify 2x2 = 32 Divide both sides by 2 x2 = 16 Solve by extracting square root x = ± √(16) = ± 4 , two solutions Solve the equation: -0.25x + 1.3 = -0.55x - 0.2 Solve the equation: -0.25x2 + 1.5 = -10.75 What is the slope of a line perpendicular to the line x = -3? What is the slope of a line parallel to the line x = 5? What is the slope of a line perpendicular to the line y = 6? Answers to the Above Questions 2.3 × 1011 22 70 18 13 6x3 -17x + 10 x3 (1/27)(x3 / y15) (1/8)(y18 / x12) 9 < M × N -3 y = -3x - 5 -1 x > 4 -4 , 4 -5 -7 , 7 0 undefined undefined Algebra Questions and problems More ACT, SAT and Compass practice