Find the value of: $\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}}$
Answer
363.9k+ views
Hint: Use the information ${i^2} = - 1$ and ${i^4} = 1$.
We know that, ${i^2} = - 1$ and ${i^4} = 1$. Using these formulae, we can write these imaginary numbers as:
$
{i^6} = {i^4} \times {i^2} = 1 \times - 1 = - 1 \\
{i^7} = {i^1} \times {i^4} \times {i^2} = i \times 1 \times - 1 = - i \\
{i^8} = {i^4} \times {i^4} = 1 \times 1 = 1 \\
{i^9} = {i^1} \times {i^4} \times {i^4} = i \times 1 \times 1 = i \\
{i^3} = {i^2} \times i = - i \\
$
Using above information in the given question,
$\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}} \Rightarrow \dfrac{{ - 1 - i + 1 + i}}{{ - 1 - i}} \Rightarrow \dfrac{0}{{ - 1 - i}} = 0$
Hence, the value of the given expression is 0.
Note: There are n number of questions which depend upon the same concept. We just need to calculate the expression in powers of i and then putting the value will give us the answer.
We know that, ${i^2} = - 1$ and ${i^4} = 1$. Using these formulae, we can write these imaginary numbers as:
$
{i^6} = {i^4} \times {i^2} = 1 \times - 1 = - 1 \\
{i^7} = {i^1} \times {i^4} \times {i^2} = i \times 1 \times - 1 = - i \\
{i^8} = {i^4} \times {i^4} = 1 \times 1 = 1 \\
{i^9} = {i^1} \times {i^4} \times {i^4} = i \times 1 \times 1 = i \\
{i^3} = {i^2} \times i = - i \\
$
Using above information in the given question,
$\dfrac{{{i^6} + {i^7} + {i^8} + {i^9}}}{{{i^2} + {i^3}}} \Rightarrow \dfrac{{ - 1 - i + 1 + i}}{{ - 1 - i}} \Rightarrow \dfrac{0}{{ - 1 - i}} = 0$
Hence, the value of the given expression is 0.
Note: There are n number of questions which depend upon the same concept. We just need to calculate the expression in powers of i and then putting the value will give us the answer.
Last updated date: 27th Sep 2023
•
Total views: 363.9k
•
Views today: 4.63k
Recently Updated Pages
What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

One cusec is equal to how many liters class 8 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE
