MHB Where Did I Go Wrong When Simplifying This Exponent Equation?

[PistolSlap]

I have this problem to simplify with positive exponents:

\[\left[(-4a^{-4}b^{-5})^{-3}\right]^4\]

So, working with the interior brackets, I applied -3 to the equation, which resulted in:

\[-(-64)x^{12}b^{15}\]

**because "-4" was not in brackets, the exponent was applied to the 4 only, independent of the negative sign, which resulted in -(-64), so:

\[\left[ 64x^{12}b^{15}\right]^4\]

which resulted in this insane answer:

\[16777216x^{48}b^{60}\]

However, when I checked it, an online calculator said the answer was:

\[\frac{a^{48}b^{60}}{16777216}\]

which means that when applying exponents to the number I did something wrong, as it should have ended up negative, which would have resulted in it becoming a positive denominator.

What did I do wrong to end up with the wrong sign on that number?

 

[MarkFL]

Re: Simplifying Question -- Please Help?

In your first step, you incorrectly applied the exponent of -3 to the constant -4, you applied a positive 3 instead. Your first step should look like:

$\displaystyle \left[\left(-4a^{-4}b^{-5} \right)^{-3} \right]^4=\left[(-4)^{1(-3)}a^{-4(-3)}b^{-5(-3)} \right]^4=\left[(-4)^{-3}a^{12}b^{15} \right]^4=\left[-\frac{a^{12}b^{15}}{4^3} \right]^4$

Now, you can see why the result is as given by the online calculator you used.

 

[Deveno]

the various negative signs make this kind of complicated.

the first thing i would do is recognize that:

$-4a^{-4}b^{-5} = (-1)(4)(a^{-4}b^{-5})$

so:

$(-4a^{-4}b^{-5})^{-3} = (-1)^{-3}(4)^{-3}(a^{-4})^{-3}(b^{-5})^{-3}$

and:

$(-1)^{-3} = \dfrac{1}{(-1)^3} = \dfrac{1}{-1} = -1$, so

$(-1)^{-3}(4)^{-3}(a^{-4})^{-3}(b^{-5})^{-3} = -[(4)^{-3}(a^{-4})^{-3}(b^{-5})^{-3}]$

now, "inside the brackets" the first term is:

$4^{-3} = \dfrac{1}{4^3}$, so we have:

$(-4a^{-3}b^{-5})^{-3} = -\left(\dfrac{a^{(-4)(-3)}b^{(-5)(-3)}}{4^3}\right) = -\left(\dfrac{a^{12}b^{15}}{4^3}\right)$

taking the 4th power of this, the negative sign goes away, and we get:

$[(-4a^{-3}b^{-5})^{-3}]^4 = \left[-\left(\dfrac{a^{12}b^{15}}{4^3}\right)\right]^4 = \left(\dfrac{a^{12}b^{15}}{4^3}\right)^4$

$ = \dfrac{a^{48}b^{60}}{4^{12}}$

as a side note, your reasoning that since (-4) was "not in brackets" the exponent did not apply to the negative sign but only to the 4 is wrong...you just got lucky, because -3 is ODD.

for example:

$(-2a)^2 = 4a^2$ but $-(2a)^2 = -4a^2$

since the first is $(-2a)(-2a)$ while the second is $-(2a)(2a)$.

 

[PistolSlap]

Re: Simplifying Question -- Please Help?

Awesome, thanks, I understand now! :D

In your first step, you incorrectly applied the exponent of -3 to the constant -4, you applied a positive 3 instead. Your first step should look like:

$\displaystyle \left[\left(-4a^{-4}b^{-5} \right)^{-3} \right]^4=\left[(-4)^{1(-3)}a^{-4(-3)}b^{-5(-3)} \right]^4=\left[(-4)^{-3}a^{12}b^{15} \right]^4=\left[-\frac{a^{12}b^{15}}{4^3} \right]^4$

Now, you can see why the result is as given by the online calculator you used.