A bacteria culture doubles every hour. Starting with 1 bacterium, how many are there after 6 hours? — 64.After n hours: 2ⁿ bacteria. After 6 hours: 2⁶ = 64.
Evaluate: (−2)⁴ and (−2)³. Which is positive? — (−2)⁴ = 16 (even exponent → positive).An even power of a negative number is positive: (−2)⁴ = 16. An odd power is negative: (−2)³ = −8.
Earth's distance from the Sun is about 1.5 × 10⁸ km. Light travels 3 × 10⁵ km/s. How many seconds does light take to reach Earth? — 5 × 10² s.Time = distance/speed = (1.5 × 10⁸)/(3 × 10⁵) = 0.5 × 10³ = 5 × 10² seconds (≈ 8 minutes).
Which is the largest number? — 1.1 × 10⁹.1.1 × 10⁹ = 1,100,000,000 is larger than 6.2 × 10⁸ = 620,000,000. Compare exponents first.
A microbe is 3 × 10⁻⁵ m wide. If 10 of them are placed end-to-end, what is the total length in scientific notation? — 3 × 10⁻⁴ m.10 × (3 × 10⁻⁵) = 30 × 10⁻⁵ = 3 × 10⁻⁴ m.
Convert the result of (1.2 × 10⁴) + (8 × 10³) to proper scientific notation. — 2.0 × 10⁴.8 × 10³ = 0.8 × 10⁴. Then (1.2 + 0.8) × 10⁴ = 2.0 × 10⁴.
Write 0.00000000089 in scientific notation. — 8.9 × 10⁻¹⁰.Count 10 places from the left of 8.9 to the original decimal position → exponent −10.
The product (3 × 10²) × (4 × 10⁴) expressed correctly is: — 1.2 × 10⁷.3 × 4 = 12 and 10² × 10⁴ = 10⁶. Then 12 × 10⁶ = 1.2 × 10⁷.
Why must the coefficient in scientific notation be between 1 and 10? — To ensure a unique, standard representation for every number.Requiring 1 ≤ |a| < 10 guarantees each number has exactly one scientific notation form, making comparison unambiguous.
A computer performs 2.6 × 10⁹ operations per second. How many operations does it perform in 3 × 10² seconds? — 7.8 × 10¹¹.2.6 × 3 = 7.8 and 10⁹ × 10² = 10¹¹. Result: 7.8 × 10¹¹ operations.
Compare: 7.0 × 10⁻⁴ and 3.5 × 10⁻³. Which is larger? — 3.5 × 10⁻³.3.5 × 10⁻³ = 0.0035 and 7.0 × 10⁻⁴ = 0.0007. So 3.5 × 10⁻³ is larger.
Solve the system: y = x + 2 and y = 3x − 4. — (3, 5).Set equal: x + 2 = 3x − 4 → 6 = 2x → x = 3. Then y = 3 + 2 = 5. Solution: (3, 5).
Solve the system: 2x + y = 8 and x − y = 1. — (3, 2).Add the equations: 3x = 9 → x = 3. Then y = 8 − 2(3) = 2. Solution: (3, 2).
Which of these systems has no solution? — y = 2x + 1 and y = 2x − 3.Parallel lines (same slope, different intercepts) never intersect: y = 2x + 1 and y = 2x − 3 both have slope 2 but different y-intercepts.
A system of two linear equations has infinitely many solutions when: — The two equations represent the same line.If both equations describe the same line, every point on that line satisfies both equations → infinitely many solutions.
Find the slope of a line parallel to y = −(1/2)x + 4. — −1/2.Parallel lines have equal slopes. The slope of y = −(1/2)x + 4 is −1/2.
What is the slope of a line perpendicular to y = 3x + 1? — −1/3.Perpendicular slopes are negative reciprocals: the negative reciprocal of 3 is −1/3.
Solve by elimination: 3x + 2y = 12 and 3x − y = 3. — (2, 3).Subtract the second from the first: 3y = 9 → y = 3. Then 3x = 12 − 6 = 6 → x = 2. Solution: (2, 3).
A line passes through (0, −2) with slope 5. What is the y-value when x = 3? — 13.y = 5(3) + (−2) = 15 − 2 = 13.
Find the x-intercept of y = 3x − 9. — (3, 0).Set y = 0: 3x − 9 = 0 → x = 3. x-intercept is (3, 0).
Write an equation in slope-intercept form for a line with slope −3 passing through (1, 4). — y = −3x + 7.Use y = mx + b: 4 = −3(1) + b → b = 7. Equation: y = −3x + 7.
The cost of a gym membership is $30/month. A competitor costs $50 setup + $20/month. After how many months is total cost the same? — 5 months.30m = 50 + 20m → 10m = 50 → m = 5 months.
A slope of −1/4 means: — For every 4 units right, the line drops 1 unit.Rise = −1, run = 4: the line falls 1 unit for every 4 units of horizontal movement to the right.