How do you square a negative exponent for kinetic energy calculation?

[Drakkith]

Trying to find out the kinetic energy that a 1kg ball gives the Earth if dropped from 100 ft. I just don't know how to square a negative exponent. 4x10^-24 squared is? I was thinking it would be 4x10^-48, but I really don't know. If that is correct, would the result be different if the exponent was an odd number? Like 4x10^-23 squared?

 

[Mentallic]

Is it $$\left(4\cdot 10^{24}\right)^2$$ or $$4\cdot \left(10^{24}\right)^2$$ ?

Either way, consider the rules $$(ab)^2=a^2\cdot b^2$$ and $$\left(a^b\right)^c=a^{bc}$$

 

[Drakkith]

Should be (4x10^-24)^2

 

[HallsofIvy]

$(a^b)^c= a^{bc}$ so you are multiplying the exponents.

(-24)(2)= -48.

$$(4\cdot 10^{-24})^2= 4^2\cdot 10^{(2)(-24)}= 16\cdot 10^{-48}$$

If you were writing this in "scientific notation" where the first number is always between 1 and 10, it would be
$$1.6\cdot 10^{-47}$$

 

[Drakkith]

Awesome. Thanks.