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lora16 [44]
3 years ago
6

What is the value of the expression i 0 × i 1 × i 2 × i 3 × i 4?

Mathematics
2 answers:
astraxan [27]3 years ago
8 0
ANSWER


The value of the expression is
- 1


EXPLANATION

Method 1: Rewrite as product of
{i}^{2}


The expression given to us is,

{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}


We use the fact that
{i}^{2}  =  - 1
to simplify the above expression.



{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  {i}^{0}  \times {i}^{1}  \times {i}^{3}   \times {i}^{2}   \times {i}^{4}


This implies,


{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  {i}^{0}  \times {i}^{2}  \times {i}^{2}   \times {i}^{2}   \times {i}^{2} \times {i}^{2}


We substitute to obtain,

{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  1\times  - 1 \times  - 1  \times  - 1\times  - 1 \times  - 1


{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  1\times  1 \times   1  \times  - 1 =  - 1


Method 2: Use indices to solve.



{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  = {i}^{0 + 1 + 2 + 3 + 4}



This implies that,


{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  = {i}^{10}




{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  (  {{i}^{2}} )^{5}


{i}^{0}  \times {i}^{1}  \times {i}^{2}  \times {i}^{3}  \times {i}^{4}  =  (   - 1 )^{5}   =  - 1


grigory [225]3 years ago
3 0

- 1

<h3>Further explanation</h3>

This is a problem that is partly related to complex numbers, i.e., imaginary numbers. We will see how the power of i is an imaginary unit. Maybe we will see an interesting pattern.

\boxed{\boxed{ \ i = \sqrt{-1}\ }} \rightarrow \boxed{\boxed{ \ i^2 = -1 \ }}

<u>Question: </u>

The value of the expression \boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 \ }

<u>The Process</u>

Recall \boxed{ \ x^0 = 1 \ }.

\boxed{ \ i^0 = 1 \ }

\boxed{ \ i^1 = \sqrt{-1} \ or \ i \ }

\boxed{ \ i^2 = (\sqrt{-1})^2 = -1 \ }

\boxed{ \ i^3 = i \times i^2 = i \times -1 = -i \ }

\boxed{ \ i^4 = i^2 \times i^2 = -1 \times -1 = 1\ }

Then \boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = 1 \times i \times (-1) \times -i \times 1 \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = -1 \times -i^2 \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = i^2 \ }

The result is  \boxed{\boxed{ \ -1 \ }}

- - - - - - -

Another method is to use the property of indices.

\boxed{ \ x^a \cdot x^b \rightleftharpoons x^{a+b} \ } \ and \ \boxed{ \ (x^a)^b) \rightleftharpoons x^{ab} \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = i^{0 + 1 + 2 + 3 + 4} \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = i^{10} \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = (i^2)^5 \ }

\boxed{ \ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = {-1}^5 \ }

We get the same result, i.e., \boxed{\boxed{ \ -1 \ }}

- - - - - - -

Well, now pay attention to the pattern.

\boxed{ \ i^0 = 1 \ }

\boxed{ \ i^1 = i \ }

\boxed{ \ i^2 = -1 \ }

\boxed{ \ i^3 = -i \ }

\boxed{ \ i^4 = 1\ }

\boxed{ \ i^5 = i^2 \times i^3 = i \ }

\boxed{ \ i^6 = i^2 \times i^4 = -1 \ }

\boxed{ \ i^7 = i^2 \times i^5 = -i \ }

Pattern repeat every 4^{th} power.

<h3>Learn more</h3>
  1. About complex numbers brainly.com/question/1658190
  2. The piecewise-defined functions brainly.com/question/9590016
  3. The composite function brainly.com/question/1691598

Keywords: what is the value of the expression, i⁰ × i¹ × i² × i³ × i⁴, imaginary number, unit, a complex number, the pattern, the property of indices

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