MAT 1215 College Algebra Exam 1 | Complete Practice
Questions & Detailed Solutions
Question 1
𝑥 −3 𝑦 2
Simplify the expression assuming 𝑥 ≠ 0and 𝑦 ≠ 0.
𝑥 −5 𝑦 −4
• A. 𝑥 2 𝑦 6
• B. 𝑥 −8 𝑦 −2
• C. 𝑥 2 𝑦 −2
• D. 𝑥 −2 𝑦 6
• Correct Answer: A
𝑥𝑎
• Detailed Rationale: Use the quotient rule for exponents =
𝑥𝑏
𝑥 −3 𝑦2
𝑥 𝑎−𝑏 : = 𝑥 −3−(−5) = 𝑥 2 , and = 𝑦 2−(−4) = 𝑦 6 . Combining
𝑥 −5 𝑦 −4
these gives 𝑥 2 𝑦 6 .
Question 2
Simplify the radical expression √75𝑥 3 .
• A. 5𝑥 √3𝑥
• B. 25𝑥 √3𝑥
• C. 15𝑥 √𝑥
• D. 5𝑥 2 √3
• Correct Answer: A
, • Detailed Rationale: Factor out the largest perfect square from
75𝑥 3 : 75𝑥 3 = 25 ⋅ 3 ⋅ 𝑥 2 ⋅ 𝑥 = (25𝑥 2 )(3𝑥 ). Taking the square
root of the square factor gives √25𝑥 2 ⋅ √3𝑥 = 5𝑥 √3𝑥.
Question 3
5
Rationalize the denominator of .
3−√2
5(3+√2)
• A.
7
5(3−√2)
• B.
7
• C. 3 + √2
15+5√2
• D.
11
• Correct Answer: A
• Detailed Rationale: Multiply numerator and denominator by the
5(3+√2)
conjugate of the denominator, which is 3 + √2: =
(3−√2)(3+√2)
5(3+√2) 5(3+√2)
= .
9−2 7
Question 4
Perform the addition of complex numbers: (3 + 4𝑖) + (2 − 5𝑖).
• A. 5 − 𝑖
• B. 5 + 9𝑖
• C. 1 + 9𝑖
• D. 6 − 20𝑖
• Correct Answer: A
, • Detailed Rationale: Combine real parts and imaginary parts:
(3 + 2) + (4𝑖 − 5𝑖) = 5 − 𝑖.
Question 5
Multiply the complex numbers: (2 − 3𝑖)(4 + 𝑖).
• A. 5 − 10𝑖
• B. 11 − 10𝑖
• C. 8 + 3𝑖
• D. 5 − 14𝑖
• Correct Answer: B
• Detailed Rationale: Use FOIL: 2(4) + 2(𝑖) − 3𝑖(4) − 3𝑖(𝑖) = 8 +
2𝑖 − 12𝑖 − 3𝑖 2 . Since 𝑖 2 = −1, this becomes 8 − 10𝑖 − 3(−1) =
8 + 3 − 10𝑖 = 11 − 10𝑖.
Question 6
Simplify (2𝑥 3 )4 .
• A. 8𝑥 7
• B. 16𝑥 12
• C. 8𝑥 12
• D. 16𝑥 7
• Correct Answer: B
• Detailed Rationale: Apply the power of a product rule (𝑎𝑏)𝑛 =
𝑎𝑛 𝑏 𝑛 and power rule (𝑥 𝑎 )𝑏 = 𝑥 𝑎𝑏 : 24 ⋅ (𝑥 3 )4 = 16𝑥 12 .
Question 7
, Evaluate the expression 27−2/3 .
• A. −9
1
• B. −
9
1
• C.
9
• D. 9
• Correct Answer: C
• Detailed Rationale: The fractional exponent indicates a root
−2 1 1
followed by a power: 27−2/3 = (271/3 ) = (3)−2 = = .
32 9
Question 8
Write the number 0.000456in scientific notation.
• A. 4.56 × 10−4
• B. 4.56 × 104
• C. 45.6 × 10−5
• D. 0.456 × 10−3
• Correct Answer: A
• Detailed Rationale: Move the decimal point 4places to the right
to obtain 4.56. Since the original number is less than 1, the
exponent is negative: 4.56 × 10−4 .
Question 9
Simplify the algebraic expression: 3(2𝑥 − 4) − 2(5𝑥 + 1).
• A. −4𝑥 − 14
Questions & Detailed Solutions
Question 1
𝑥 −3 𝑦 2
Simplify the expression assuming 𝑥 ≠ 0and 𝑦 ≠ 0.
𝑥 −5 𝑦 −4
• A. 𝑥 2 𝑦 6
• B. 𝑥 −8 𝑦 −2
• C. 𝑥 2 𝑦 −2
• D. 𝑥 −2 𝑦 6
• Correct Answer: A
𝑥𝑎
• Detailed Rationale: Use the quotient rule for exponents =
𝑥𝑏
𝑥 −3 𝑦2
𝑥 𝑎−𝑏 : = 𝑥 −3−(−5) = 𝑥 2 , and = 𝑦 2−(−4) = 𝑦 6 . Combining
𝑥 −5 𝑦 −4
these gives 𝑥 2 𝑦 6 .
Question 2
Simplify the radical expression √75𝑥 3 .
• A. 5𝑥 √3𝑥
• B. 25𝑥 √3𝑥
• C. 15𝑥 √𝑥
• D. 5𝑥 2 √3
• Correct Answer: A
, • Detailed Rationale: Factor out the largest perfect square from
75𝑥 3 : 75𝑥 3 = 25 ⋅ 3 ⋅ 𝑥 2 ⋅ 𝑥 = (25𝑥 2 )(3𝑥 ). Taking the square
root of the square factor gives √25𝑥 2 ⋅ √3𝑥 = 5𝑥 √3𝑥.
Question 3
5
Rationalize the denominator of .
3−√2
5(3+√2)
• A.
7
5(3−√2)
• B.
7
• C. 3 + √2
15+5√2
• D.
11
• Correct Answer: A
• Detailed Rationale: Multiply numerator and denominator by the
5(3+√2)
conjugate of the denominator, which is 3 + √2: =
(3−√2)(3+√2)
5(3+√2) 5(3+√2)
= .
9−2 7
Question 4
Perform the addition of complex numbers: (3 + 4𝑖) + (2 − 5𝑖).
• A. 5 − 𝑖
• B. 5 + 9𝑖
• C. 1 + 9𝑖
• D. 6 − 20𝑖
• Correct Answer: A
, • Detailed Rationale: Combine real parts and imaginary parts:
(3 + 2) + (4𝑖 − 5𝑖) = 5 − 𝑖.
Question 5
Multiply the complex numbers: (2 − 3𝑖)(4 + 𝑖).
• A. 5 − 10𝑖
• B. 11 − 10𝑖
• C. 8 + 3𝑖
• D. 5 − 14𝑖
• Correct Answer: B
• Detailed Rationale: Use FOIL: 2(4) + 2(𝑖) − 3𝑖(4) − 3𝑖(𝑖) = 8 +
2𝑖 − 12𝑖 − 3𝑖 2 . Since 𝑖 2 = −1, this becomes 8 − 10𝑖 − 3(−1) =
8 + 3 − 10𝑖 = 11 − 10𝑖.
Question 6
Simplify (2𝑥 3 )4 .
• A. 8𝑥 7
• B. 16𝑥 12
• C. 8𝑥 12
• D. 16𝑥 7
• Correct Answer: B
• Detailed Rationale: Apply the power of a product rule (𝑎𝑏)𝑛 =
𝑎𝑛 𝑏 𝑛 and power rule (𝑥 𝑎 )𝑏 = 𝑥 𝑎𝑏 : 24 ⋅ (𝑥 3 )4 = 16𝑥 12 .
Question 7
, Evaluate the expression 27−2/3 .
• A. −9
1
• B. −
9
1
• C.
9
• D. 9
• Correct Answer: C
• Detailed Rationale: The fractional exponent indicates a root
−2 1 1
followed by a power: 27−2/3 = (271/3 ) = (3)−2 = = .
32 9
Question 8
Write the number 0.000456in scientific notation.
• A. 4.56 × 10−4
• B. 4.56 × 104
• C. 45.6 × 10−5
• D. 0.456 × 10−3
• Correct Answer: A
• Detailed Rationale: Move the decimal point 4places to the right
to obtain 4.56. Since the original number is less than 1, the
exponent is negative: 4.56 × 10−4 .
Question 9
Simplify the algebraic expression: 3(2𝑥 − 4) − 2(5𝑥 + 1).
• A. −4𝑥 − 14
