What is an expression to represent One minus the quotient of one and a number x.

Recall the binomial theorem.

$(a+b)^n = \displaystyle \sum_{k=0}^n \binom nk a^{n-k} b^k$

1. The binomial expansion of $\left(1+\frac x3\right)^7$ is

$\left(1 + \dfrac x3\right)^7 = \displaystyle\sum_{k=0}^7 \binom 7k 1^{7-k} \left(\frac x3\right)^k = \sum_{k=0}^7 \binom 7k \frac{x^k}{3^k}$

Observe that

$k = 1 \implies \dbinom 71 \left(\dfrac x3\right)^1 = \dfrac73 x$

$k = 2 \implies \dbinom 72 \left(\dfrac x3\right)^2 = \dfrac73 x^2$

When we multiply these by $8-9x$ ,

• $8$ and $\frac73 x^2$ combine to make $\frac{56}3 x^2$

• $-9x$ and $\frac73 x$ combine to make $-\frac{63}3 x^2 = -21x^2$

and the sum of these terms is

$\dfrac{56}3 x^2 - 21x^2 = \boxed{-\dfrac73 x^2}$

2. The binomial expansion is

$\left(2a - \dfrac b2\right)^8 = \displaystyle \sum_{k=0}^8 \binom 8k (2a)^{8-k} \left(-\frac b2\right)^k = \sum_{k=0}^8 \binom 8k 2^{8-2k} a^{8-k} b^k$

We get the $a^6b^2$ term when $k=2$ :

$k=2 \implies \dbinom 82 2^{8-2\cdot2} a^{8-2} b^2 = 28 \cdot2^4 a^6 b^2 = \boxed{448} \, a^6b^2$

6 0

1 year ago

How do i get a better grade if i'm failing

stellarik [79]

Ask for extra credit.

4 0

2 years ago

Read 2 more answers

Step by step explanation need it

nasty-shy [4]

Answer:

8/17

Step-by-step explanation:

The sine of an angle is defined as the opposite side to the reference angle divided by the hypotenuse.

The side opposite the angle is always the side not connected to the reference angle. In this case the opposite side = ZY

The hypotenuse = XZ

Sin(X) = ZY/XZ

Sin(X) = 1634 = 8 / 17

4 0

2 years ago

All about simulitious equations

Korolek [52]

Answer:

On occasions you will come across two or more unknown quantities, and two or more equations

relating them. These are called simultaneous equations and when asked to solve them you

must find values of the unknowns which satisfy all the given equations at the same time.

Step-by-step explanation:

1. The solution of a pair of simultaneous equations

The solution of the pair of simultaneous equations

3x + 2y = 36, and 5x + 4y = 64

is x = 8 and y = 6. This is easily verified by substituting these values into the left-hand sides

to obtain the values on the right. So x = 8, y = 6 satisfy the simultaneous equations.

2. Solving a pair of simultaneous equations

There are many ways of solving simultaneous equations. Perhaps the simplest way is elimination. This is a process which involves removing or eliminating one of the unknowns to leave a

single equation which involves the other unknown. The method is best illustrated by example.

Example

Solve the simultaneous equations 3x + 2y = 36 (1)

5x + 4y = 64 (2) .

Solution

Notice that if we multiply both sides of the first equation by 2 we obtain an equivalent equation

6x + 4y = 72 (3)

Now, if equation (2) is subtracted from equation (3) the terms involving y will be eliminated:

6x + 4y = 72 − (3)

5x + 4y = 64 (2)

x + 0y = 8

5 0

2 years ago

5700 is compounded semiannually at a rate of 15% for 12 years

Rama09 [41]

Answer:

$32,335.38

Step-by-step explanation:

You are going to want to use the compound interest formula, which is shown below.

$P(1+\frac{r}{n} )^{nt}$

P = initial balance

r = interest rate

n = number of times compounded annually

t = time

Now lets plug in the values into the equation:

$5,700(1+\frac{.15}{2})^{12(2)}$ = 32,335.38

Your answer is $32,335.38

4 0

2 years ago