Question

simplify the complex numbers: -10/5-7i

How do you know to conjugate this problem because of the -7i in the denominator?

Answer 1

\[\large \frac{-10}{5-7i}\]

`How do you know to... `

Yes, it's because we have an imaginary, or complex rather, value in the denominator. So we want to get that out of there.

So what is the conjugate of 5-7i? :D

Answer 2

Thanks I want to make sure I know this info for tomorrow! 5+7i/5t7i

Answer 3

Safe to say when it states simplify the complex number and it's in the denominator I use conjugate correct?

Answer 4

Yah, we don't want to leave complex numbers in the denominator c:
The same thing comes up when you deal with square roots.

We don't leave `irrational` numbers in the denominator if we can avoid it.
Example:\[\large \frac{1}{\sqrt 2} \qquad = \qquad \frac{2}{\sqrt 2}\]
We remove the irrational number \(\large \sqrt2\) from the denominator using some multiplication.

Blah I got off topic a bit there :D lol

Answer 5

Woops, I wrote that backwards T.T\[\large \frac{1}{\sqrt 2} \qquad = \qquad \frac{\sqrt2}{ 2}\]

Answer 6

do you mind breaking down the hook part of the square root sign and how I could make it into a fraction we could use example \[\sqrt[3]{8x^3}\]

Answer 7

We can rewrite a root as a `rational expression` by using a fraction in our exponent, as you mentioned.

Example:\[\large \sqrt {x^3} \qquad = \qquad x^{3/2}\]

The numerator of our fraction represents the POWER on x.
The denominator represents the degree of the root.

When no number is shown in front of the root, it signifies the `square` root, or second root. As I'm sure you're aware.

Answer 8

With the example you gave, we don't really need to use this idea.
We first recognize that 8 is a `perfect cube`.
8=2*2*2.

\[\large \sqrt[3]{8x^3} \qquad = \qquad \sqrt[3]{2^3x^3}\qquad = \qquad \sqrt[3]{(2x)^3}\]

Answer 9

Right what if instead of 8 place 7 would that work in making a rational expression

Answer 10

Sure, let's try that.\[\large \sqrt[3]{7x^3} \qquad = \qquad \left(7x^3\right)^{1/3}\]

We can rewrite the 3rd root as 1/3.
The one power (no change), and a 3 in the denominator letting us know it's a third root.

From here, we need to remember our rules for exponents.

Answer 11

One important rule is: We have to apply the exponent to EACH VALUE inside.\[\large \left(7x^3\right)^{1/3} \qquad = \qquad 7^{1/3}(x^3)^{1/3}\]

Answer 12

Another rule: When you have a term being raised to a power, raised to a power, you `multiply` the exponents.
So we simply multiply the 3 and 1/3.\[\large 7^{1/3}(x^3)^{1/3} \qquad = \qquad 7^{1/3}x^{3/3}\qquad = \qquad 7^{1/3}x\]

Answer 13

Do you mind showing me a few more examples on how to make rational expressions when given them in square root form

Answer 14

Let's try this one.

\[\large \sqrt{4x^3}\]

Answer 15

so it's (4x)^2/3?

Answer 16

Maybe it will make more sense if we don't rationalize the root right away. I have a feeling that step is confusing you. Let's try simplifying first.\[\large \sqrt{4x^3} \qquad = \qquad \sqrt4 \cdot \sqrt{x^3}\]All I did was split up the roots so far.

Answer 17

Okay

Answer 18

Sqrt4=?

Answer 19

Got it so far 2

Answer 20

\[\large \sqrt4 \cdot \sqrt{x^3} \qquad = \qquad 2 \sqrt{x^3}\]
Ok good. So from here, we'll apply this idea of writing it as a rational expression.
It's a second root (our denominator will be a 2) and a 3 power (our numerator will be a 3).

Answer 21

so is it x^2/3?

Answer 22

Where did your 2 in front go? :D
Don't forget about that 2!

Answer 23

2x*^2/3

Answer 24

2x^2/3

Answer 25

3 was our power. It should be on TOP of the fraction.

Answer 26

the 2 was the degree of the root, DENOMINATOR :D

Answer 27

\[\large 2 \sqrt{x^3} \qquad = \qquad 2x^{3/2}\]

Answer 28

when its x^n inside the square root its the denominator?