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
FromTheMoon [43]
3 years ago
14

Prove that the statement (ab)^n=a^n * b^n is true using mathematical induction.

Mathematics
2 answers:
IRISSAK [1]3 years ago
6 0

Hello, n being an integer, we need to prove that one statement depending on n is true, let's note it S(n).

The mathematical induction involves two steps:

Step 1 - We need to prove S(1), meaning that the statement is true for n = 1

Step 2 - for k integer > 1, we assume S(k) and we need to prove that S(k+1) is true.

Imagine that you are a painter and you need to paint all the trees on one side of a road. You have several colours that you can use but you are asked to follow two rules:

Rule 1 - You need to paint the first tree in white.

Rule 2 - If one tree is white you have to paint the next one in white too.

What colour do you think all the trees will be painted?

Do you see why this is very important to prove the two steps as well ?

Let's do it in this example.

Step 1 - for n = 1, let's prove that S(1) is true, meaning  (ab)^1=a\cdot b =a^1\cdot b^1

So the statement is true for n = 1

Step 2 - Let's assume that this is true for k, and we have to prove that this is true for k+1

So we assume S(k), meaning that (ab)^k=a^k\cdot b^k

and what about S(k+1), meaning (ab)^{k+1}=a^{k+1}\cdot b^{k+1} ?

We will use the fact that this is true for k,

(ab)^{k+1}=(ab)\cdot (ab)^k =(ab) \cdot a^k \cdot b^k

We can write it because the statement at k is true and then we can conclude.

(ab)^{k+1}=(ab)\cdot (ab)^k =(ab) \cdot a^k \cdot b^k=a^{k+1}\cdot b^{k+1}

In conclusion, we have just proved that S(n) is true for any n integer greater or equal to 1, meaning (ab)^{n}=a^{n}\cdot b^{n}

Hope this helps.

Do not hesitate if you need further explanation.

Thank you

mamaluj [8]3 years ago
5 0

Answer:

see below

Step-by-step explanation:

      (ab)^n=a^n * b^n

We need to show that it is true for n=1

assuming that it is true for n = k;

(ab)^n=a^n * b^n

( ab) ^1 = a^1 * b^1

ab = a * b

ab = ab

Then we need to show that it is true for n = ( k+1)

or (ab)^(k+1)=a^( k+1) * b^( k+1)

Starting with

  (ab)^k=a^k * b^k    given

Multiply each side by ab

ab *  (ab)^k= ab *a^k * b^k

   ( ab) ^ ( k+1) = a^ ( k+1) b^ (k+1)

Therefore, the rule is true for every natural number n

You might be interested in
Rechel is paid $11 for every hour at her dog walking job she is also get tip.she earned a total of $73 for 6 hours of work.how m
Jet001 [13]

Answer:

So you would think 6 hours= 11$ so 6x11=66 now 73-66= 7 so her tip was 7 dollars

please mark me brainliest

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Coterminal angle between 0 degrees and 360 degrees if coterminal angle is -450
aliina [53]

Answer:

uhh

Step-by-step explanation:

the answ free points

4 0
2 years ago
Please answer the question in the photo
Inessa05 [86]

Answer: Your answer is C.

Step-by-step explanation: The number 35 is an irrational number because it can’t be expressed as a ratio. And the other are a bit confusing for me to explain why your answer is C but I’m very positive that your answer is C

3 0
3 years ago
20.3 is 4 times as large as some number. What is the number?
neonofarm [45]
The answer would be 81.2 because all you gotta do it 20.3 times 4

7 0
3 years ago
Read 2 more answers
Complete the remainder of the<br> table for the given function rule:
melamori03 [73]

Answer:

-3=2

0=0

3=-2

6=-4

3 0
3 years ago
Other questions:
  • Ms. Gartland bought x number of shirts for the new members of her chorus. The cost for the x number of shirts, including $3.99 s
    6·1 answer
  • The height in feet of a rocket launched into the air is modeled by the function (t)=-16t^2+160t+300 where t is time in seconds.
    11·1 answer
  • A 4   cm × 6   cm 4cm×6cmstart color #1fab54, 4, start text, c, m, end text, times, 6, start text, c, m, end text, end color #1f
    15·2 answers
  • What is the population in orlando fl<br> also who live in bradenton
    7·2 answers
  • Given PQRS, what is the measure of Q
    10·2 answers
  • From the diagram below, you can say that the two triangles are
    14·1 answer
  • Which of the following relations between input numbers and output numbers is not a function?
    12·1 answer
  • Rate of change for coordinates (8,-2) (0,4) (-4,7)
    5·1 answer
  • The head of maintenance at XYZ Rent-A-Car believes that the mean number of miles between services is 3408 miles, with a variance
    8·1 answer
  • Simplify the expression.<br> 3 - 4 +7 - 4
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!