QuestionSimplify $i^{10}$ using exponent laws. Write it as $\left(i^{4}\right)^{[?]} i$ and find the multiples of 4.

Studdy Solution

STEP 1

Assumptions1. $i$ is the imaginary unit, which is defined as $i = \sqrt{-1}$ . . The laws of exponents apply to imaginary numbers as well.

STEP 2

We know that $i^{4} = (i^{2})^{2} = (-1)^{2} =1$ . This is a key property of the imaginary unit that we will use to simplify the expression.

STEP 3

We can express $i^{10}$ as $\left(i^{}\right)^{[?]} i$ . To find the number of powers with a multiple of, we divide10 by.
$\frac{10}{} =2 \text{ remainder }2$ This tells us that we can express $i^{10}$ as $\left(i^{}\right)^{2} i^{2}$ .

STEP 4

Now we can simplify the expression using the property of $i^{4}$ that we stated in2.
$\left(i^{4}\right)^{2} i^{2} =1^{2} i^{2} = i^{2}$

STEP 5

Finally, we simplify $i^{2}$ using the definition of the imaginary unit.
$i^{2} = -1$ So, $i^{10} = -1$ .

Was this helpful?

QuestionSimplify $i^{10}$ using exponent laws. Write it as $\left(i^{4}\right)^{[?]} i$ and find the multiples of 4.

Studdy Solution

STEP 1

Assumptions1. $i$ is the imaginary unit, which is defined as $i = \sqrt{-1}$ . . The laws of exponents apply to imaginary numbers as well.

STEP 2

We know that $i^{4} = (i^{2})^{2} = (-1)^{2} =1$ . This is a key property of the imaginary unit that we will use to simplify the expression.

STEP 3

We can express $i^{10}$ as $\left(i^{}\right)^{[?]} i$ . To find the number of powers with a multiple of, we divide10 by.
$\frac{10}{} =2 \text{ remainder }2$ This tells us that we can express $i^{10}$ as $\left(i^{}\right)^{2} i^{2}$ .

STEP 4

Now we can simplify the expression using the property of $i^{4}$ that we stated in2.
$\left(i^{4}\right)^{2} i^{2} =1^{2} i^{2} = i^{2}$

STEP 5

Finally, we simplify $i^{2}$ using the definition of the imaginary unit.
$i^{2} = -1$ So, $i^{10} = -1$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSimplify i10i^{10}i10 using exponent laws. Write it as (i4)[?]i\left(i^{4}\right)^{[?]} i(i4)[?]i and find the multiples of 4.

Studdy Solution

STEP 1

Assumptions1. iii is the imaginary unit, which is defined as i=−1i = \sqrt{-1}i=−1​. . The laws of exponents apply to imaginary numbers as well.

STEP 2

We know that i4=(i2)2=(−1)2=1i^{4} = (i^{2})^{2} = (-1)^{2} =1i4=(i2)2=(−1)2=1. This is a key property of the imaginary unit that we will use to simplify the expression.

STEP 3

We can express i10i^{10}i10 as (i)[?]i\left(i^{}\right)^{[?]} i(i)[?]i. To find the number of powers with a multiple of, we divide10 by.10=2 remainder 2\frac{10}{} =2 \text{ remainder }210​=2 remainder 2This tells us that we can express i10i^{10}i10 as (i)2i2\left(i^{}\right)^{2} i^{2}(i)2i2.

STEP 4

Now we can simplify the expression using the property of i4i^{4}i4 that we stated in2.(i4)2i2=12i2=i2\left(i^{4}\right)^{2} i^{2} =1^{2} i^{2} = i^{2}(i4)2i2=12i2=i2

STEP 5

Finally, we simplify i2i^{2}i2 using the definition of the imaginary unit.i2=−1i^{2} = -1i2=−1So, i10=−1i^{10} = -1i10=−1.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSimplify $i^{10}$ using exponent laws. Write it as $\left(i^{4}\right)^{[?]} i$ and find the multiples of 4.

Assumptions1. $i$ is the imaginary unit, which is defined as $i = \sqrt{-1}$ . . The laws of exponents apply to imaginary numbers as well.

We know that $i^{4} = (i^{2})^{2} = (-1)^{2} =1$ . This is a key property of the imaginary unit that we will use to simplify the expression.

We can express $i^{10}$ as $\left(i^{}\right)^{[?]} i$ . To find the number of powers with a multiple of, we divide10 by.
$\frac{10}{} =2 \text{ remainder }2$ This tells us that we can express $i^{10}$ as $\left(i^{}\right)^{2} i^{2}$ .

Now we can simplify the expression using the property of $i^{4}$ that we stated in2.
$\left(i^{4}\right)^{2} i^{2} =1^{2} i^{2} = i^{2}$

Finally, we simplify $i^{2}$ using the definition of the imaginary unit.
$i^{2} = -1$ So, $i^{10} = -1$ .