Simplify the root: -18^ sqrt(-8)^18 (i get -18 ^ 8^9)

[Panama]

Problem:
-18^ ?(-8)^18 =

Does this answer look right?
-18 ^ 8^9

 

[stapel]

Problem: -18^ ?(-8)^18 =

Assuming the instructions were to simplify, is the original expression as follows:

. . . . .\(\displaystyle -18^{\sqrt{(-8)^{18}}}\)

...or is it something else?

Thank you!

Eliz.

 

[Panama]

The -18 on the left is suppose to be the index /root. But everything is is correct.

Index..> -18^ sqrt(-8)^18

 

[Loren]

Are you saying

\(\displaystyle -\sqrt[18]{-8}^{18}\)

or \(\displaystyle \sqrt[-18]{-8}^{18}\)

or \(\displaystyle -\sqrt[18]{(-8)^{18}}\)

or something else?

 

[Panama]

The last one is the equation.

 

[mmm4444bot]

\(\displaystyle -\sqrt[2]{(-8)^{2}}\;=\;-\sqrt[2]{(-8)(-8)}\;=\;-\sqrt[2]{8 \cdot 8}\;=\;-\sqrt[2]{64}\;=\;-8\)

\(\displaystyle -\sqrt[4]{(-8)^{4}}\;=\;-\sqrt[4]{(-8)(-8)(-8)(-8)}\;=\;-\sqrt[4]{8 \cdot 8 \cdot 8 \cdot 8}\;=\;-\sqrt[4]{4096}\;=\;-8\)

\(\displaystyle -\sqrt[6]{(-8)^{6}}\;=\;-\sqrt[6]{(-8)(-8)(-8)(-8)(-8)(-8)}\;=\;-\sqrt[6]{8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8}\;=\;-\sqrt[6]{262144}\;=\;-8\)

\(\displaystyle -\sqrt[8]{(-8)^{8}}\;=\;-\sqrt[8]{(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)}\;=\;-\sqrt[8]{8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8}\;=\;-\sqrt[8]{16777216}\;=\;-8\)

\(\displaystyle -\sqrt[10]{(-8)^{10}}\;=\;-\sqrt[10]{(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)}\;=\;-\sqrt[10]{8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8}\;=\;-\sqrt[10]{1073741824}\;=\;-8\)

\(\displaystyle -\sqrt[12]{(-8)^{12}}\;=\;-\sqrt[12]{(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)(-8)}\;=\;-\sqrt[12]{8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8}\;=\;-\sqrt[12]{68719476736}\;=\;-8\)



Do you notice a pattern?

 

[Panama]

The answer always becomes a -8? Is that the pattern you were speaking of?

 

[Denis]

Panama, what is [sqrt(x)]^2 ?
If you don't know that, you shouldn't even attempt this problem!

 

[mmm4444bot]

The answer always becomes a -8?

Not always. Only when the index of the radical is both even and it matches the exponent on the radicand.

EG

\(\displaystyle - \sqrt[3]{(-8)^3} \; = \; - \sqrt[3]{(-8)(-8)(-8)} \; = \; - \sqrt[3]{-512} \; = \; -(-8) \; = \; 8\)