Given that, in standard form, `3^236` is approx. `4 * 10 ^ 112` and `3^(-376)` is approx. `4 * 10^(-180),` find the approximation in standard form...

Hello!

To answer this question we only need the fact  `3^(-376) approx 4*10^(-180).`

By the definition, raise some number `b` to a negative natural power `-n` means 1) raise `b` to the positive power `n` and 2) divide `1` by the result. This is the formula:

`b^(-n) = 1/(b^n).`

As you can easily infer from this formula,

`b^(n) = 1/(b^(-n))`                              ...

Hello!

To answer this question we only need the fact  `3^(-376) approx 4*10^(-180).`

By the definition, raise some number `b` to a negative natural power `-n` means 1) raise `b` to the positive power `n` and 2) divide `1` by the result. This is the formula:

`b^(-n) = 1/(b^n).`

As you can easily infer from this formula,

`b^(n) = 1/(b^(-n))`                                                     (1)

is also true.

In our task, `n = 376` and `b = 3.` So we have

`3^376 = 1/(3^(-376)).`

The number at the denominator is approximately known, so

`3^376 = 1/(3^(-376)) approx 1/(4*10^(-180)) = 1/4*1/(10^(-180)) = 0.25*1/(10^(-180)).`

Now we use the formula (1) in the reverse direction for `b = 10` and `n = 180:`

`1/10^(-180) = 10^180.`

This way the number in question is about

`0.25*10^180 = 0.25*10*10^179 = 2.5*10^179`  

(standard form requires factor between `1` and `10` ).

So the answer is:  `3^376 approx 2.5*10^179.`

(if you actually need 3 in some other degree, please reply and I'll try to help)