University:
Jackson College
Course:
MAT-010 | Basic Mathematics
Academic year:

2024
Views:

131

Pages:

2
Author:

Bennett Wheeler

Question 1 Express the fraction 1 6 4 6 4 1 using a negative exponent. Answer 1 Using the negative exponent rule 𝑎 − 𝑛 = 1 𝑎 𝑛 a −n = a n 1 : 1 6 4 = 6 − 4 6 4 1 =6 −4 So, the answer is 6 − 4 6 −4 . Question 2 According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities? Answer 1 Let 𝑥 x be the number of years spent sleeping, and 𝑦 y be the number of years spent watching TV. We are given: 𝑥 + 𝑦 = 37 (total time spent sleeping and watching TV) x+y=37(total time spent sleeping and watching TV) 𝑥 = 𝑦 + 19 (sleeping exceeds TV watching by 19 years) x=y+19(sleeping exceeds TV watching by 19 years) Substitute 𝑥 = 𝑦 + 19 x=y+19 from the second equation into the first equation: ( 𝑦 + 19 ) + 𝑦 = 37 (y+19)+y=37 2 𝑦 + 19 = 37 2y+19=37 Solve for 𝑦 y: 2 𝑦 = 18 2y=18 𝑦 = 9 y=9 Substitute 𝑦 = 9 y=9 back into 𝑥 = 𝑦 + 19 x=y+19: 𝑥 = 9 + 19 = 28 x=9+19=28 So, you will spend 28 years sleeping and 9 years watching TV. Question 3 Solve the inequality: 4 𝑣 + 9 ≤ 5 4v+9≤5 or − 3 𝑣 ≥ − 6 −3v≥−6. Answer 1 Solve the first inequality 4 𝑣 + 9 ≤ 5 4v+9≤5: 4 𝑣 ≤ − 4 4v≤−4 𝑣 ≤ − 1 v≤−1 Solve the second inequality − 3 𝑣 ≥ − 6 −3v≥−6: Divide by -3, remembering to reverse the inequality sign: 𝑣 ≤ 2 v≤2 Since this is an "or" compound inequality, the solution is the union of 𝑣 ≤ − 1 v≤−1 and 𝑣 ≤ 2 v≤2, which covers all values 𝑣 ≤ 2 v≤2. So, the solution is 𝑣 ≤ 2 v≤2. Question 4 Simplify − 54 −54 using the imaginary number 𝑖 i. Choices: A) 3 𝑖 6 3i 6 B) − 3 𝑖 6 −3i 6 C) 𝑖 54 i 54 D) 3 − 6 3 −6 Answer 1 Rewrite − 54 −54 as 𝑖 54 i 54 , since − 𝑎 = 𝑖 𝑎 −a =i a for 𝑎 > 0 a>0. Simplify 54 54 by factoring: 54 = 9 ⋅ 6 = 3 6 54 = 9⋅6 =3 6 Substitute back: − 54 = 𝑖 ⋅ 3 6 = 3 𝑖 6 −54 =i⋅3 6 =3i 6 So, the answer is A) 3 𝑖 6 3i 6 . Question 5 Find the value of the expression 2 𝑎 0 + 𝑥 1 3 2a 0 + 3 x 1 when 𝑥 = 125 x=125. Answer 1 Substitute 𝑥 = 125 x=125 into the expression and simplify using 𝑎 0 = 1 a 0 =1 for any non-zero 𝑎 a: 2 𝑎 0 + 12 5 1 3 2a 0 + 3 125 1 = 2 ( 1 ) + 125 3 =2(1)+ 3 125 Simplify: = 2 + 125 3 = 2 + 41.6667 =2+ 3 125 =2+41.6667 = 43.6667 =43.6667 So, the answer is 43.6667 43.6667 (or approximately 43.67).