Mulitiply. Simplify your answer.
1. 3r^2t/6st^3 x 2r^2s^3t^2/8r^4s^2
Add or subtract. Simplify your answer.
2. x^2 + x/x^2 + 3x + 2 + 3/x + 2
3. The Escobar family went on a car trip. They drove 100 miles on country roads and 240 miles on the highway. They drove 50% faster on the highway than on the country roads. Let r represent their rate on country roads in miles per hour.
a. Write and simplify an expression that represents the number of hours it took the Escobar family to complete their trip in terms of r.
b. Find their total travel time if they drove the posted speed limit.
Please!
by now these should be getting easier. When multiplying/dividing powers, add/subtract the exponents.
3r^2t/6st^3 x 2r^2s^3t^2/8r^4s^2
= (3*2)/(6*8) r^(2+2-4) s^(-1+3-2) t^(-3+1+2)
= 1/8
You gotta start using parentheses...
x^2 + x/x^2 + 3x + 2 + 3/x + 2
= (x^2+x)/(x^2+3x+2) + 3/(x+2)
= x(x+1) / (x+2)(x+1) + 3/(x+2)
= x/(x+2) + 3/(x+2)
= (x+3)/(x+2)
Since time = distance/speed,
h = 100/r + 240/(1.5r)
Now you have to know what the posted speed limits were.
A jogger on a treadmill is jogging at 5 miles per hour. Which of the following is the time it will take him to job 1 mile?
The posted speed limit is 40mph.
To simplify the given expressions, we will follow the steps below:
1. For multiplication of rational expressions:
- Multiply the numerators together.
- Multiply the denominators together.
- Simplify if possible.
2. For addition and subtraction of rational expressions:
- Find a common denominator.
- Add or subtract the numerators.
- Simplify if possible.
Let's simplify the given expressions step by step:
1. 3r^2t/6st^3 x 2r^2s^3t^2/8r^4s^2
Multiplying the numerators and denominators together, we get:
(3r^2t) * (2r^2s^3t^2) / (6st^3) * (8r^4s^2)
Multiplying the terms within each numerator and denominator, we get:
(6r^4s^4t^3) / (48r^4s^3t^3)
To simplify further, we can divide both the numerator and denominator by their greatest common factor, which is 6r^4s^3t^3:
(6r^4s^4t^3) / (48r^4s^3t^3) = (r^4s^4) / (8)
Therefore, the simplified answer is r^4s^4/8.
2. x^2 + x / x^2 + 3x + 2 + 3 / x + 2
To simplify this expression, we need to find a common denominator for the fractions.
The common denominator for x^2 + x and x^2 + 3x + 2 is (x + 2).
Now, we can rewrite the expression with the common denominator:
[(x^2 + x)*(x + 2)] / (x^2 + 3x + 2) + (3*(x + 2)) / (x + 2)
Expanding the numerator in the first fraction and canceling out (x + 2) terms in the second fraction, we get:
[(x^3 + 2x^2 + x^2 + 2x) + (3x + 6)] / (x^2 + 3x + 2)
Combining like terms, we have:
(x^3 + 3x^2 + 4x + 6) / (x^2 + 3x + 2)
This cannot be simplified any further, so the answer is (x^3 + 3x^2 + 4x + 6) / (x^2 + 3x + 2).
3. The Escobar family went on a car trip. They drove 100 miles on country roads and 240 miles on the highway. They drove 50% faster on the highway than on the country roads. Let r represent their rate on country roads in miles per hour.
a. To find the number of hours it took the Escobar family to complete their trip in terms of r, we need to calculate the time taken for each segment and add them together.
Time taken on country roads = Distance / Rate = 100 / r hours
Time taken on the highway = Distance / Rate = 240 / (1.5r) hours (as they drove 50% faster, their rate on the highway is 1.5 times their rate on country roads)
Total time taken = Time on country roads + Time on the highway = (100 / r) + (240 / (1.5r))
To simplify this expression, we can divide both terms by the greatest common factor, which is r:
Total time taken = (100 / r) + (240 / (1.5r)) = (100 + 160) / r = 260 / r
Therefore, the expression representing the number of hours it took the Escobar family to complete their trip in terms of r is 260 / r.
b. If they drove the posted speed limit, we can assume their rate on both country roads and the highway is equal to the posted speed limit.
Let's denote the posted speed limit as s.
Time taken on country roads = Distance / Rate = 100 / s hours
Time taken on the highway = Distance / Rate = 240 / s hours (as they drove the same speed on the highway as on country roads)
Total time taken = Time on country roads + Time on the highway = (100 / s) + (240 / s)
Combining like terms, we have:
Total time taken = (100 + 240) / s = 340 / s
Therefore, if they drove the posted speed limit, their total travel time would be 340 / s.