1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Lerok [7]
3 years ago
14

Use the binomial theorem to expand the following binomial expressions.

Mathematics
1 answer:
andre [41]3 years ago
5 0

Answer:

Remember, the expansion of (x+y)^n is (x+y)^n=\sum_{k=0}^n \binom{n}{k}x^{n-k}y^k, where \binom{n}{k}=\frac{n!}{(n-k)!k!}.

a)

(1+\sqrt{2})^5=\sum_{k=0}^5 \binom{5}{k}1^{5-k}\sqrt{2}^k=\sum_{k=0}^5 \binom{5}{k}2^{\frac{k}{2}}\\=\binom{5}{0}2^{\frac{0}{2}}+\binom{5}{1}2^{\frac{1}{2}}+\binom{5}{2}2^{\frac{2}{2}}+\binom{5}{3}2^{\frac{3}{2}}+\binom{5}{4}2^{\frac{4}{2}}+\binom{5}{5}2^{\frac{5}{2}}\\=1+5\sqrt{2}+10*2+10*2^{\frac{3}{2}}+5*4+1*2^{\frac{5}{2}}\\=41+5\sqrt{2}+10*2^{\frac{3}{2}}+2^{\frac{5}{2}}

b)

(1+i)^9=\sum_{k=0}^9 \binom{9}{k}1^{9-k}i^k=\sum_{k=0}^9 \binom{9}{k}i^k\\=\binom{9}{0}i^0+\binom{9}{1}i^1+\binom{9}{2}i^2+\binom{9}{3}i^3+\binom{9}{4}i^4+\binom{9}{5}i^5+\binom{9}{6}i^6+\\+\binom{9}{7}i^7+\binom{9}{8}i^8+\binom{9}{9}i^9\\=1+9i-36-84i+126+126i-+84-36i+9+i\\=16+16i

c)

(1-\pi)^5=\sum_{k=0}^5 \binom{5}{k}1^{9-k}(-\pi)^k=\sum_{k=0}^5 \binom{5}{k}(-\pi)^k\\=\binom{5}{0}(-\pi)^0+\binom{5}{1}(-\pi)^1+\binom{5}{2}(-\pi)^2+\binom{5}{3}(-\pi)^3+\binom{5}{4}(-\pi)^4+\binom{5}{5}(-\pi)^5\\=1-5+10\pi^2-10\pi^3+5\pi^4-\pi^5

d)

(\sqrt{2}+i)^6=\sum_{k=0}^6 \binom{6}{k}\sqrt{2}^{6-k}i^k\\=\binom{6}{0}\sqrt{2}^{6}i^0+\binom{6}{1}\sqrt{2}^{5}i+\binom{6}{2}\sqrt{2}^{4}i^2+\binom{6}{3}\sqrt{2}^{3}i^3+\binom{6}{4}\sqrt{2}^{2}i^4+\binom{6}{5}\sqrt{2}i^5+\binom{6}{6}\sqrt{2}^{0}i^6

e)

(2-i)^6=\sum_{k=0}^6 \binom{6}{k}2^{6-k}(-i)^k\\=\binom{6}{0}2^{6}(-i)^0+\binom{6}{1}2^{5}(-i)^1+\binom{6}{2}2^{4}(-i)^2+\binom{6}{3}2^{3}(-i)^3+\binom{6}{4}2^{2}(-i)^4+\binom{6}{5}2^{1}(-i)^5+\binom{6}{k}2^{0}(-i)^6\\=1-32i+\binom{6}{2}16i^2-\binom{6}{3}8i^3+\binom{6}{4}4i^4-\binom{6}{5}2i^5+\binom{6}{k}i^6

You might be interested in
30 pointss!HELP! Two students get into an argument about the correct answer.
Mamont248 [21]

Both Rhianna and Christopher were correct as they took different reference angles.

Step-by-step explanation:

Step 1:

tan \theta = \frac{opposite side}{adjacentside} .

The value of the tan of a reference angle is calculated using the above formula.

The hypotenuse is the longest side of the triangle is always opposite the right angle of the triangle.

In this triangle, the hypotenuse is the side measuring 5 units.

Step 2:

When the reference angle is A, the opposite side measures 3 units and the adjacent side measures 4 units.

So tan A = \frac{3}{4} .

When the reference angle is C, the opposite side measures 4 units and the adjacent side measures 3 units.

So tan C = \frac{4}{3} .

So both Rhianna and Christopher were right as they took different reference angles i.e. A and C.

7 0
3 years ago
Shania bought a $1455 drum set on the installment plan. The installment agreement included a 15% down payment and 18 monthly pay
ahrayia [7]

Answer:

A

Step-by-step explanation:

3 0
2 years ago
Solve the equation 5x-2x^2+1
Pani-rosa [81]

Answer:

Step-by-step explanation:

If you call "5x-2x^2+1" an "equation," then you must equate 5x-2x^2+1 to 0:

5x-2x^2+1 = 0

This is a quadratic equation.  Rearranging the terms in descending order by powers of x, we get:

-2x^2 + 5x + 1 = 0.  Here the coefficients are a = -2, b = 5 and c = 1.

Use the quadratic formula to solve for x:

First find the discriminant, b^2 - 4ac:  25 - 4(-2)(1) = 25 + 8 = 33

Because the discriminant is positive, the roots of this quadratic are real and unequal.

                                                             -b ± √(discriminant)

Applying the quadratic formula   x = --------------------------------

                                                                         2a

we get:

      -5 ± √33           -5 + √33

x = ----------------- = --------------------- and

           2(-2)                     -4

                                  -5 - √33

                                 ---------------

                                         -4

5 0
3 years ago
Find the value of the variable that results in congruent triangles
sdas [7]

Answer:

  y = 5

Step-by-step explanation:

The triangles will be congruent when corresponding sides are congruent. __

We already have ...

  KL ≅ NP . . . = 15 mm

  JL ≅ MP . . . = 22 mm

So, we need ...

  JK ≅ MN

  4y +12 = 32 . . . . substitute given values

This will be the case when ...

  4y = 20 . . . . . subtract 12

  y = 5 . . . . . . . divide by 4

The value of y that makes the triangles congruent is 5.

8 0
2 years ago
The highest common factor of 18, 24, 36 is
yawa3891 [41]
If you list all the common factors, (1,2,3,6), you see the highest common factor is 6
6 0
3 years ago
Read 2 more answers
Other questions:
  • What is the equation of the line -3x-2y=30 in slope intercept form?
    9·2 answers
  • What is the equation of the parabola with vertex (-2,5) and an x-intercept of x = 3
    10·1 answer
  • You have to simplify it <br> -2x+3-(5-6x)
    9·2 answers
  • What is the solution to –4|–2x + 6| = –24?x = 0x = 0 or x = –6x = 0 or x = 6no solution
    14·1 answer
  • What is the value of x in the figure shown below?
    10·1 answer
  • Can someone help me figure this one out please?
    11·1 answer
  • Helpl please urgent!! plzzzzzzzzzzzz
    5·1 answer
  • Hellp me please DDDDDDDDDDD:
    15·2 answers
  • At a hockey game, a vender sold a combined total of 235 sodas and hot dogs. The number of hot dogs sold was 59 less than the num
    5·1 answer
  • Solve for q.mn + q = pq = p - mnq = p (mn )q = p + mn
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!