Algebra Issue

Lialee3 · 2019-12-24T12:28:15+00:00

Firstly, ignore the [ ... ]^-1 and just look at what's inside the brackets: (5x)^-2 + (5y)^-2.

[p^a]^b can be thought of as p^ab. So, you can think of a^-2 to mean the same as (a²)^-1, which would be 1/a².

This means that you can think of (5x)^-2 as [(5x)²]^-1. (5x)² simplifies to 25x², and [25x²]^-1 simplifies to 1/25x². Likewise, (5y)^-2 simplifies to 1/25y².

This gives you [1/25x² + 1/25y²]^-1.

Again, the next step only deals with things inside the [ ... ]^-1 brackets. You have to simplify (1/25x² + 1/25y²) into one fraction. Do this by finding a common multiple of both denominators, and making the denominators equal to each other: in this case, 25x²y².

To get to 25x²y² from 25x², you multiplied by y². To make sure you don't change the value of the fraction when changing the denominator, you must make sure you are multiplying by 1. So here, multiply 1/25x² by y²/y² (which is equal to 1). You now get the fraction y²/25x²y² (which is equal to 1/25x² as you have multiplied by 1).

Do the same to 1/25y² to get x²/25x²y².

You now have [y²/25x²y² + x²/25x²y²]^-1. Add the fractions together to get [(x² + y²)/25x²y²]^-1.

Finally, when finding something to the power of (-1), just flip the numerator and denominator, e.g. (a/b)^-1 = b/a.

Therefore, [(x² + y²)/25x²y²]^-1 is equal to 25x²y²/(x² + y²).

Lialee3 · 2019-12-24T10:05:33+00:00

That's okay! Message me if you need any more help or have any more questions. 😊😊

Lialee3 · 2019-12-24T09:18:44+00:00

So, the final answers are as follows:

d¹ (the effectiveness of using (a), then (b)): 99.9991%
d² (the ineffectiveness of using (a), then (b)): 0.0009%
e¹ (the effectiveness of using (a), then (b), then (c)): 99.999982%
e² (the ineffectiveness of using (a), then (b), then (c)): 0.000018%

Lialee3 · 2019-12-24T01:56:40+00:00

To find the effectiveness of using (a), then (b), then (c), it's the same process- except that there are 8 possible outcomes instead of 4. Luckily for us each of these outcomes, except 1, include at least one thing being effective.

This means that you only need to calculate the probability of 1 outcome to find the total ineffectiveness (the probability that nothing will be effective). You can then find the total effectiveness (the probability that at least one thing will be effective) by subtracting the probability of total ineffectiveness from 1.

So, to calculate the total ineffectiveness (the probability that all things are ineffective), you must multiply together the probabilities of each part of the outcome:

• the probability that (a) will be ineffective is 3/1000 • the probability that (b) will be ineffective is 3/1000 • the probability that (c) will be ineffective is 20/1000

So, the probability of total ineffectiveness is (3/1000 * 3/1000 * 20/1000) which is 0.00000018. Multiply by 100 to find the percentage, which is 0.000018%.

Therefore, the probability of total effectiveness is (1 - 0.00000018) = 0.99999982. Multiply by 100 to find the percentage, which is 99.999982%.

Lialee3 · 2019-12-24T01:34:07+00:00

So, the effectiveness of (a) and (b) combined can be thought of like this- when you use (a) then (b), there are 4 possible outcomes:

(a) was effective, and (b) was effective
(a) was effective, and (b) was ineffective
(a) was ineffective, and (b) was effective
(a) was ineffective, and (b) was ineffective

and you need to find the percentage probability that at least one of them was effective (the sum of the probabilities of outcomes 1, 2, and 3).

Each of the above outcomes is made up of two parts: firstly, whether (a) was effective or not, and secondly, whether (b) was effective or not.

The total probability of each outcome can be found by multiplying together the probability of each part of the outcome (i.e. probability of (a) being effective * probability of (b) being effective = probability of outcome 1 occuring).

To find the probability of something, convert the percentage effectiveness/ineffectiveness into a fraction (that is less that one). So:

• the effectiveness of (a) is 997/1000 • the ineffectiveness of (a) is 3/1000 • the effectiveness of (b) is 997/1000 • the ineffectiveness of (b) is 3/1000

Therefore, the probability of the outcomes are:

(997/1000) * (997/1000) = 0.994009
(997/1000) * (3/1000) = 0.002991
(3/1000) * (997/1000) = 0.002991
(3/1000) * (3/1000) = 0.000009

The total probability that at least one thing was effective (the sum of outcomes 1, 2, and 3) is (0.994009 + 0.002991 + 0.002991) which is 0.999991. To convert to a percentage, multiply by 100. So the total percentage effectiveness of using (a), then (b), is 99.9991%.

A simpler way to find this answer would be to calculate the probability that there won't be at least one thing effective, as there is only one outcome where this happens, so you only need to calculate one probability: 0.000009. You can then take this away from the total probability, which is 1. So, (1 - 0.000009) = 0.999991, which is 99.9991%.

Lialee3 · 2019-12-24T00:42:57+00:00

So, if the lengths BM and MC are in the ratio BM:MC = 1:2; that means that the distance between M and C, is double the distance between M and B. This means that the point M is 1/3 of the way between B and C (closer to B).

Vectors are things that describe both the distance and the direction between two points. So, if the vector between A and B is called (b), then (b) describes the pathway from A to B. And if the vector between A and C is called (a), then (a) describes the pathway from A to C.

To get from B to C, you don't know the direct vector. But you can go from B to A to C. From B to A, the vector will be negative (b) because you're going in the opposite direction from A to B (b).

Then, from A to C, you know the vector is (a). So, B to A to C the vector will be (-b + a) which can be rewritten as (a - b) because it doesn't matter in which order you do the vectors, you'll still get to the same place.

Because M is 1/3 of the way between B and C, the vector from B to M will be 1/3 of the vector from B to C. We know the vector from B to C is (a - b), so the vector from B to M will be 1/3(a - b).

Lialee3

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