Question 1 Write each complex number in standard form. 2 ( cos ⁡ 3 0 ∘ + 𝑖 sin ⁡ 3 0 ∘ ) 2(cos30 ∘ +isin30 ∘ ) Answer 1 The standard form of a complex number is 𝑎 + 𝑏 𝑖 a+bi. Given: 2 ( cos ⁡ 3 0 ∘ + 𝑖 sin ⁡ 3 0 ∘ ) 2(cos30 ∘ +isin30 ∘ ) Substitute cos ⁡ 3 0 ∘ = 3 2 cos30 ∘ = 2 3 and sin ⁡ 3 0 ∘ = 1 2 sin30 ∘ = 2 1 : 2 ( 3 2 + 𝑖 1 2 ) = 3 + 𝑖 2( 2 3 +i 2 1 )= 3 +i So, the standard form of the given complex number is 3 + 𝑖 3 +i. Question 2 Which statement is correct? 3.56 ⋅ 1 0 2 1.09 ⋅ 1 0 4 or 3.56 ⋅ 1 0 2 ≤ ( 4.08 ⋅ 1 0 2 ) ( 1.95 ⋅ 1 0 − 6 ) 1.09⋅10 4 3.56⋅10 2 or3.56⋅10 2 ≤(4.08⋅10 2 )(1.95⋅10 −6 ) Answer 1 First, evaluate 3.56 ⋅ 1 0 2 1.09 ⋅ 1 0 4 1.09⋅10 4 3.56⋅10 2 : = 356 10900 = 0.0326 = 3.26 × 1 0 − 2 = 10900 356 =0.0326=3.26×10 −2 Next, evaluate ( 4.08 ⋅ 1 0 2 ) ( 1.95 ⋅ 1 0 − 6 ) (4.08⋅10 2 )(1.95⋅10 −6 ): = ( 4.08 ) ( 1.95 ) ⋅ 1 0 2 − 6 = 7.956 ⋅ 1 0 − 4 =(4.08)(1.95)⋅10 2−6 =7.956⋅10 −4 Comparing, we find: 3.56 ⋅ 1 0 2 < ( 4.08 ⋅ 1 0 2 ) ( 1.95 ⋅ 1 0 − 6 ) 3.56⋅10 2 <(4.08⋅10 2 )(1.95⋅10 −6 ) Thus, the correct statement is: 3.56 ⋅ 1 0 2 1.09 ⋅ 1 0 4 < ( 4.08 ⋅ 1 0 2 ) ( 1.95 ⋅ 1 0 − 6 ) 1.09⋅10 4 3.56⋅10 2 <(4.08⋅10 2 )(1.95⋅10 −6 ) Question 3 A credit card contains 16 digits between 0 and 9. However, only 100 million numbers are valid. If a number is entered randomly, what is the probability that it is a valid number? Answer 1 There are 1 0 16 10 16 possible 16-digit combinations (each digit has 10 choices). There are 100 100 million valid numbers, which is 100 ⋅ 1 0 6 = 1 0 8 100⋅10 6 =10 8 . Probability of a valid number: valid numbers total possible numbers = 1 0 8 1 0 16 = 1 1 0 8 total possible numbers valid numbers = 10 16 10 8 = 10 8 1 The probability is 1 1 0 8 10 8 1 , which is very low. Question 4 Write ( − 5 ) × ⋯ × ( − 5 ) (−5)×⋯×(−5) (17 times) in exponential form. Answer 1 A number multiplied by itself 𝑛 n times can be written as 𝑎 𝑛 a n , where 𝑎 a is the base and 𝑛 n is the exponent. ( − 5 ) × ( − 5 ) × ⋯ × ( − 5 ) = ( − 5 ) 17 (−5)×(−5)×⋯×(−5)=(−5) 17 Question 5 The exponential expression 2 8 2 8 has a value of 256. Write two other exponential expressions that have a value of 256. Answer 1 2 8 = 256 2 8 =256 Using the associative property of multiplication: ( 2 ⋅ 2 ) ⋅ ( 2 ⋅ 2 ) ⋅ ( 2 ⋅ 2 ) ⋅ ( 2 ⋅ 2 ) = 4 4 = 256 (2⋅2)⋅(2⋅2)⋅(2⋅2)⋅(2⋅2)=4 4 =256 ( 2 ⋅ 2 ⋅ 2 ⋅ 2 ) ⋅ ( 2 ⋅ 2 ⋅ 2 ⋅ 2 ) = 1 6 2 = 256 (2⋅2⋅2⋅2)⋅(2⋅2⋅2⋅2)=16 2 =256 So, the two expressions are 4 4 4 4 and 1 6 2 16 2 .