Simplifying Question -- Please Help?

[PistolSlap]

I have this problem to simplify with positive exponents:

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

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

which resulted in this insane answer:

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

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?

 

[Deleted member 4993]

I have this problem to simplify with positive exponents:

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

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

which resulted in this insane answer:

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

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?

\(\displaystyle 4^{-3} \ = \ \dfrac{1}{64}\)

 

[JeffM]

I have this problem to simplify with positive exponents:

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

where did the x come from

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

if the exponent was applied to the 4 independently of the sign, why is 4 cubed - 64?

so:

which resulted in this insane answer:

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

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?

I like to get rid of negative exponents before doing anything else. (This is not a logical requirement; I just make fewer careless errors that way.)

So \(\displaystyle \left\{\left(-4a^{-4}b^{-5}\right)^{-3}\right\}^4 = \left\{\left(- 4 * \dfrac{1}{a^4} * \dfrac{1}{b^5}\right)^{-3}\right\}^4 = \left\{\left(- \dfrac{4}{a^4b^5}\right)^{-3}\right\}^4.\)

So that got rid of the negative exponents inside the innermost brackets. But we still have a negative exponent to get rid of.

\(\displaystyle \left\{\left(-4a^{-4}b^{-5}\right)^{-3}\right\}^4 = \left\{\left(- \dfrac{4}{a^4b^5}\right)^{-3}\right\}^4 = \left\{\left(-\dfrac{a^4b^5}{4}\right)^3\right\}^4.\)

Another way I make silly mistakes is exponentiating negative numbers so I separate out the negative as follows

\(\displaystyle \left\{\left(-4a^{-4}b^{-5}\right)^{-3}\right\}^4 = \left\{\left(-\dfrac{a^4b^5}{4}\right)^3\right\}^4 = \left\{\left(-1\right)^3\right\}^4 * \left\{\left(\dfrac{a^4b^5}{4}\right)^3\right\}^4.\)

Now it's easy.

\(\displaystyle \left\{\left(-4a^{-4}b^{-5}\right)^{-3}\right\}^4 = \left\{\left(-1\right)^3\right\}^4 * \left\{\left(\dfrac{a^4b^5}{4}\right)^3\right\}^4 = (-1)^{(3 * 4)} * \left(\dfrac{a^4b^5}{4}\right)^{(3 * 4)} = (-1)^{12} * \dfrac{a^{(4 * 12)}b^{(5 * 12)}}{4^{12}} = 1 * \dfrac{a^{48}b^{60}}{16,777,216} = \dfrac{a^{48}b^{60}}{16,777,216}.\)

Just take it one step at a time

 

[HallsofIvy]

I have this problem to simplify with positive exponents:

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

What happened to the "-"? What happened to the "a" and where did "x" come from? Are you really paying attention to what you are doing? (A problem I always had.) \(\displaystyle (-4)^{-3}=-\frac{1}{64}\). If you are going to put a "-" inside the third power, you cannot also have one outside. \(\displaystyle (a^{-4})^{-3}= a^{12}\). \(\displaystyle (b^{-5})^{-3}= b^{15}\).

So \(\displaystyle (-4 a^{-4}b^{-5})^{-3}= -\frac{a^{12}b^{15}}{64}\).

Now take that to the fourth power: \(\displaystyle (64)^4= 16777216\). (Yes, I used a calculator for that.) Any negative number to a fourth (even) power is positive so the "-" is gone. \(\displaystyle (a^{12})^4= a^{4(12)}= a^{48}\). \(\displaystyle (b^{15})^4= b^{4(15)}= b^{60}\).

So \(\displaystyle [(-4a^{-4}b^{-5})^{-3}]^4= \frac{a^{48}b^{60}}{16777216}\).

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

,
?? The "-4" certainly is inside the "( )" to which the exponent -3 is applied.

so:

which resulted in this insane answer:

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

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?

I have no idea what you are saying. In the first place, while the answer you got is wrong, I don't see why you say "insane". You also say "it should have ended up negative". What should have? Are you talking about the fact that "-4" is negative? That is NOT an exponent so has nothing to do with whether it is in the numerator or denominator, only with whether the result is itself positive or negative. And any (real) number to an even power, like 4, is non-negative.