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In the expansion (ax+by)^7, the coefficients of the first two terms are 128 and -224, respectively. Find the values of a and b

madam [21]

Answer:

a = 2, b = 3.5

Step-by-step explanation:

Expanding $(ax+by)^7$ using Binomial expansion, we have that:

$(ax+by)^7$ =

$(ax)^7(by)^0 + (ax)^6(by)^1 + (ax)^5(by)^2 + (ax)^4(by)^3 + (ax)^3(by)^4 + (ax)^2(by)^5 + (ax)^1(by)^6 + (ax)^0(by)^7$

$= (a)^7(x)^7+ (a)^6(x)^6(b)(y) + (a)^5(x)^5(b)^2(y)^2 + (a)^4(x)^4(b)^3(y)^3 + (a)^3(x)^3(b)^4(y)^4 + (a)^2(x)^2(b)^5(y)^5 + (a)(x)(b)^6(y)^6 + (b)^7(y)^7\\\\\\= (a)^7(x)^7+ (a)^6(b)(x)^6(y) + (a)^5(b)^2(x)^5(y)^2 + (a)^4(b)^3(x)^4(y)^3 + (a)^3(b)^4(x)^3(y)^4 + (a)^2(b)^5(x)^2(y)^5 + (a)(b)^6(x)(y)^6 + (b)^7(y)^7$

We have that the coefficients of the first two terms are 128 and -224.

For the first term:

=> $a^7 = 128$

=> $a = \sqrt[7]{128}\\ \\\\a = 2$

For the second term:

$a^6b = -224$

$b = \frac{-224}{a^6}$

$b = \frac{-224}{2^6} \\\\\\b = \frac{-224}{64} \\\\\\b = 3.5$

Therefore, a = 2, b = 3.5

5 0

2 years ago

SOMEONE PLEASE ANSWER AND I WILL GIVE BRAINLIEST :))A company makes poppy lapel pins. Each of them costs £1.10 to make. In Octob

Liono4ka [1.6K]

Answer: The company will make 720 dollars (I dont know the percentage out of that though).

Step-by-step explanation: 1.10 times 1000= 1100. 35 percent of that is 385 dollars worth to charity or 350 lapel pins. 1000 minus 350 equals 650. 650 times .6 equals 390. 390 times 3 dollars= 1170 dollars. 650 times .4 = 260. 260 divided by 2 = 130, 130 times 5 dollars equals 650 dollars. 650 plus 1170= 1820. 1820-1100= 720.

3 0

2 years ago

Discuss four ways to keep the nervous system healthy

geniusboy [140]

Exercise regularly.

Do not smoke or use tobacco products.

Get plenty of rest.

Eat a balanced diet.

7 0

2 years ago

Read 2 more answers

Your friend asks you to help him study and will pay you 5 dimes the first time you help him. You agree to help if he multiplies

Viefleur [7K]

Answer:

78125 dimes

Step-by-step explanation:

The payment for sessions makes a series

5 dimes, 5*5 = 25 dimes, 25*5 = 125 dimes, ...

This series is GP with the first term of 5 and common ratio of 5.

The nth term would be

$t_n=5*5^{n-1}$

Use the equation above and find the 7th term

$t_7=5*5^6 = 5^7 = 78125$

6 0

1 year ago

Ken normally leaves work at 5:00 pm, but he is leaving 20 minutes late today. He decides to make up time by taking the toll road

kobusy [5.1K]

Answer:

The correct answer is: $y=\frac{1}{3} x + 20$

Step-by-step explanation:

As stated in the question that x represents the number of minutes his normal commute takes. Here the keyword is normal; in the normal commute, Ken takes side streets instead of the toll road.

However, in this scenario, we have to come up with the equation that takes Ken's commute via toll road.

Ken can travel 3 times faster by taking the toll road (Given), which can be represented in the mathematical terms in terms of x as follows:

$x_{toll\_road}=\frac{1}{3}x$

Without considering Ken getting late, the equation would become the following:

$y=\frac{1}{3} x$ --- (A)

As Ken is leaving late, we have to incorporate that time as well by adding it in the aforementioned equation (A).

In this case, it's 20 minutes; therefore, the equation (A) will become:

$y=\frac{1}{3} x+20$

Hence, the correct answer is $y=\frac{1}{3} x + 20$ .

4 0

2 years ago

Who want a turturer?only 5 student!​

Who want a turturer?only 5 student!