Question

solve and simplify this expression.

Answer 1

\[\left( -\frac{ 2 }{ 3 } \right)^{-4}\]

Answer 2

It's booooook.

Answer 3

Didn't you just ask this one a sec ago? :d
Still confused?

Answer 4

yeah. I was just getting frustrated with the others. it's been a long day.

Answer 5

To switch your exponent from negative to positive,
you flip the fraction,\[\large\rm \left(\frac{a}{b}\right)^{-c}\quad=\quad \left(\frac{b}{a}\right)^{+c}\]

Answer 6

Another example:\[\large\rm \left(-\frac{4}{7}\right)^{-2}\quad=\quad \left(-\frac{7}{4}\right)^2\]

Answer 7

then you can distribute and the fraction still has the negative sign in front?

Answer 8

Whether or not we still have a negative,
completely depends on the `parity`of our exponent (odd or even).\[\large\rm \left(-\frac{2}{3}\right)^{-4}\quad=\quad\left(-\frac{3}{2}\right)^4\]
Even exponents will cause the negative to disappear because an even number of negatives multiplied together becomes positive, ya? :)\[\large\rm -^2=(-)(-)=(+)\]

Answer 9

yeah. but because we flipped it to make the exponent positive, the fraction would stay negative right?

Answer 10

Yes, good call,
`the flipping` had no effect on the negative.
Keep in mind we could pair the negative with either the 2 or the 3,
or leave it in front as it is,\[\large\rm \left(-\frac{2}{3}\right)^{-4}\quad=\quad\left(\frac{-2}{3}\right)^{-4}\quad=\quad\left(\frac{2}{-3}\right)^{-4}\]So if we flip any of these,
we can just move the negative wherever we want again.

Answer 11

Distribute as you said,\[\large\rm \left(-\frac{3}{2}\right)^4\quad=\quad -^4\frac{3^4}{2^2}\]Maybe pair the negative with one of the values so it's less confusing notation.\[\large\rm \left(\frac{-3}{2}\right)^4\quad=\quad \frac{(-3)^4}{2^2}\]

Answer 12

yeah that looks like a good idea. I'd rather not melt my brain. :)

Answer 13

Is that as far as it goes unless you want to use larger numbers?

Answer 14

Woah that denominator is 2^4,
bad typo sorry

Answer 15

Realize that since we have an `even power`,
the negative disappears.\[\large\rm \left(\frac{-3}{2}\right)^4\quad=\quad \frac{(-3)^4}{2^4}\quad=\quad \frac{3^4}{2^4}\]Powers of 2 and 3 aren't too bad.
They won't be big big numbers :)
Should probably expand it out.

Answer 16

3*3 =9
9*3 =27
27*3=81

2*2 =4
4*2 =8
8*2 =16

Answer 17

My teacher has never said anything about being able to drop a negative sign in this situation is the exponent is a negative. Is the rule just that simple? but only if the exponent is an even number.

Answer 18

Correct :) only if even.
For this problem, let's pair them up like this,\[\large\rm (-3)^4\quad=\quad \color{orangered}{(-3)(-3)}\cdot\color{royalblue}{(-3)(-3)}\]-3*-3 gives us positive 9 in each case,\[\large\rm (-3)^4\quad=\quad \color{orangered}{(9)}\cdot\color{royalblue}{(9)}\]

Answer 19

An even number is just a bunch of `pairs`,
and a pair of negatives gives us a positive, ya?

Answer 20

wow. It didn't make much sense until I saw that visual but I get it completely now!

Answer 21

Cool!
I gotta go make some chicken.
Keep up the good work \c:/