Which of the following expressions is equivalent to: 6x³ + 7x² + 1/x?
- A. 63 + 72 + 1/x
- B. 63 + 72 + 1
- C. 6x² + 7x + 1/x
- D. 6x² + 7x + 1
- E. 6x² + 7x² + 1
Correct Answer & Rationale
Correct Answer: C
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
The expression 6x³ + 7x² + 1/x can be simplified by factoring out the highest degree of x and rearranging the terms. Option C, 6x² + 7x + 1/x, contains the correct coefficients for the x terms, but with the degrees adjusted appropriately. Option A incorrectly suggests a constant sum of 63 and 72, which does not relate to the original expression. Option B also misrepresents the original expression by omitting the variable terms entirely. Option D fails to maintain the degree of x in the cubic term, while option E mistakenly combines the x² terms incorrectly, resulting in an inaccurate expression.
Other Related Questions
When Henry plays the songs on the playlist in a random order, what is the probability a rock song will be played first?
- A. 3/4
- B. 1/3
- C. 1/4
- D. 3/10
- E. 5/16
Correct Answer & Rationale
Correct Answer: D
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
Which of the following expressions is equivalent to: 1200 × (5 × 10â·)?
- A. 12×10¹â°
- B. 6.0×10¹â°
- C. 6.0×10¹¹
- D. 7.2×10¹³
- E. 9.4×10¹â´
Correct Answer & Rationale
Correct Answer: B
To find an equivalent expression for \( 1200 \times (5 \times 10^n) \), we first simplify \( 1200 \) as \( 1.2 \times 10^3 \). Thus, the expression becomes \( 1.2 \times 10^3 \times 5 \times 10^n = 6.0 \times 10^{3+n} \). Option A incorrectly simplifies the coefficient and exponent. Option C miscalculates the exponent, not aligning with the original multiplication. Option D has an incorrect coefficient and exponent combination. Option E also miscalculates the coefficient and exponent. Therefore, only option B accurately reflects the simplified expression.
To find an equivalent expression for \( 1200 \times (5 \times 10^n) \), we first simplify \( 1200 \) as \( 1.2 \times 10^3 \). Thus, the expression becomes \( 1.2 \times 10^3 \times 5 \times 10^n = 6.0 \times 10^{3+n} \). Option A incorrectly simplifies the coefficient and exponent. Option C miscalculates the exponent, not aligning with the original multiplication. Option D has an incorrect coefficient and exponent combination. Option E also miscalculates the coefficient and exponent. Therefore, only option B accurately reflects the simplified expression.
A campground rents canoes for either $20 per day or $4 per hour. For what number or numbers of hours, h, is it more expensive to rent a canoe at the daily rate than at the hourly rate?
- A. h = 5
- B. h >= 25
- C. h > 5
- D. h < 5
- E. h ≤ 5
Correct Answer & Rationale
Correct Answer: C
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
To determine when renting a canoe at the daily rate exceeds the hourly rate, we compare the costs. The daily rate is $20, while the hourly rate is $4 per hour. Setting up the inequality, we have: \[ 20 > 4h \] Dividing both sides by 4 gives: \[ 5 > h \] This means that renting for more than 5 hours makes the daily rate more economical. Option A (h = 5) is incorrect since at 5 hours, both rates are equal. Option B (h ≥ 25) is incorrect because it's not relevant to the threshold we calculated. Option D (h < 5) suggests a scenario where the daily rate is not more expensive, which contradicts our findings. Option E (h ≤ 5) includes values where the rates are equal or less, which doesn't satisfy the condition.
A medium-sized grain of sand can be approximated as a cube with an edge length of 5×10â»â´ meters. Which expression best represents the number of medium-sized sand grains that could be lined up side by side to result in a total length of 1 meter?
- A. 2×10³
- B. 2×10â´
- C. 2×10âµ
- D. 5×10³
- E. 5×10â´
Correct Answer & Rationale
Correct Answer: B
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.