Math

I'm sorry but I'll probably at explaining this,

-4 is first increased by 3 powers. An even number of powers causes the result to be even, and in the same regard, an odd exponent on a negative number will give a negative number.

-4 x -4 = 16 (-4^2)
16 x -4 = -64 (-4^3)

Second, the result of -4 to the power of 3 is increased by the power of 2, multiplying the first result (-64) with itself, causing the second result to be even.

(-4^3)^2 = -64^2 = 4096

So, then, ((-4)^3)^2 = 4^6 = 4096

>>370367

(-4)^3)^2 = -4^(3*2) = -4^6, your solution is correct.
But since 6 is an even number a base with negative sign becomes positive, so you can leave it out. That doesn't make you solution wrong, both works!

-4^6 = 4^6

>>370401
- 4^6 = -4096
4^6 = 4096

That's what my calculator is telling me at least.

Your convulsion was (-4)^6 not -4^6. Those brackets are important.

>>370403
I see. Then I don't understand why the book said the answer was 4^6?

>>370415
because If a negative number is raised to an even power, the result will be positive, therefore
(-X)^2 = X^2
(-4)^6 = (4)^6

>>370415
because you need to write the result with a positive base according to your assignment.

Just keep in mind
(-x)^even exponent = x^even exponent
(-x)^odd exponent = - x^odd exponent

since you can extract the sign:
(-x)^n = ((-1)(x))^n = (-1)^n x^n and (-1)^n is -1 for odd n and +1 for even n.

>>370429
The book gave ((-5)^3)^-5 = -5 ^-15 as an example. I think that's what throwing me off.

Isn't -5 a negative base or am I going insane?

>>370432
no, -5^-15 means -(5^-15): positive base
That's different from (-5)^-15: negative base
even if the results are the same in this case.

-5^2 = -(5^2) = -25 for example, and not
-5^2 ?=? (-5)^2 = 25.

>>370402
>(-4)^6=4^6

>>370432

3*-5 is odd, therefore the result keeps the sign. Look at (-1)^3 = -1*-1*-1 = 1*-1 = -1
That's why the minus before the base is kept. Also in this case that is not optional like in even case because -5^-15 =|= 5^-15
Only the one with the minus is correct, also indeed keep in mind that "-a^b" with brackets means (-a)^b, you just tend to leave them out writing like this.