Math  /  Algebra

QuestionCalculate 61{ }^{61}. 61=(4)i15i1=1=\begin{aligned} 61 & =(4) i^{15} \cdot i^{1} \\ & =1 \cdot \square \\ & =\square \end{aligned} Hint: 61=461=4
Calculate 83{ }^{83} 83=(4)? Hint: 83=2=1=\begin{aligned} 83 & =(4) ? \cdot \quad \text { Hint: } 83=2 \\ & =1 \cdot \square \\ & =\square \end{aligned}

Studdy Solution

STEP 1

What is this asking? We need to calculate ii raised to the powers of **61** and **83**, and simplify the results. Watch out! Remember that powers of ii cycle through four values: i1=ii^1 = i, i2=1i^2 = -1, i3=ii^3 = -i, and i4=1i^4 = 1.
Don't get them mixed up!

STEP 2

1. Calculate i61i^{61}
2. Calculate i83i^{83}

STEP 3

Let's **rewrite** the exponent 61 as a multiple of 4 plus a remainder.
Why? Because i4=1i^4 = 1, and multiplying by 1 is super easy!
We can write 6161 as 415+14 \cdot 15 + 1.
This is just like dividing 61 by 4: we get 15 with a remainder of 1.

STEP 4

Now we can **rewrite** i61i^{61} as i415+1i^{4 \cdot 15 + 1}.
This is the same as (i4)15i1(i^4)^{15} \cdot i^1.
See how we used the properties of exponents there?

STEP 5

Since i4=1i^4 = 1, we can **substitute** that in: (1)15i1(1)^{15} \cdot i^1.
Anything raised to the power of 1 is just itself, so i1=ii^1 = i.

STEP 6

Now we have 115i1^{15} \cdot i.
One raised to any power is still just **one**, so we have 1i1 \cdot i.

STEP 7

Finally, 1i=i1 \cdot i = i.
So, i61=ii^{61} = i.
Awesome!

STEP 8

Just like before, let's **rewrite** the exponent 83 as a multiple of 4 plus a remainder.
We can write 83 as 420+34 \cdot 20 + 3.

STEP 9

Now we **rewrite** i83i^{83} as i420+3i^{4 \cdot 20 + 3}.
This is the same as (i4)20i3(i^4)^{20} \cdot i^3.

STEP 10

We know i4=1i^4 = 1, so we **substitute** that in: (1)20i3(1)^{20} \cdot i^3.

STEP 11

We have 120i31^{20} \cdot i^3.
One to any power is still just **one**, so we have 1i31 \cdot i^3.

STEP 12

Remember that i3=ii^3 = -i, so we can **substitute** that in: 1(i)1 \cdot (-i).

STEP 13

Finally, 1(i)=i1 \cdot (-i) = -i.
So, i83=ii^{83} = -i.
Fantastic!

STEP 14

i61=ii^{61} = i i83=ii^{83} = -i

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