Powers of numbers containing negative numbers

Question

I just came across this question and thought if i could ask help. How do you solve problems that have powers with a negative number? Ex. 2^(-2)

Answer 1

The effect of the minus sign in the exponent is to invert the remaining expression. For example, $2^{-3}=\frac1{2^3}$. Since $2^3$ is $8$, that means $2^{-3}=\frac1{2^3}=\frac18$.

The justification is that this preserves the ordinary laws of exponents, such as $$x^a\cdot x^b=x^{a+b}$$ regardless of the sign of $a$ or $b$.

Addenda: One can assume the above law holds, and derive the desired result (and others).

For example, to determine what $x^0$ must mean, one could proceeds as follows.

$$x=x^1=x^{0+1}=\boxed{x^0}\cdot x^1=\boxed{x^0}\cdot x$$ That is, $$x=\boxed{x^0}\cdot x$$ Dividing both sides by $x$ yields that $x^0$ must be $1$ if it means anything.

Then it follows that $$1=x^0=x^{-n+n}=\boxed{x^{-n}}\cdot x^n$$ That is, $$1=\boxed{x^{-n}}\cdot x^n$$ and dividing by $x^n$ gives that $$\boxed{x^{-n}}= \frac{1}{x^n}$$