Alan

1.  Use the fact that log_bx - log_by = log_b(x/y)

log₅180 -  log₅75 - log₅60 = log₅(1/25) → log₅(25^-1) → log₅(5^-2)

2.  Now use the fact that log_b(b^m) = m

log₅(5^-2) = -2

So log₅180 -  log₅75 - log₅60 = -2

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Use Snell's law:   sin(i)/sin(r) = n

i is angle of incidence, r is angle of refraction, so r = sin^-1(sin(35°)/1.33) ≈ 19.2°

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-13 is the correct mathematical result, because multiplication takes priority over addition and subtraction.  Scientific calculators should do this correctly.  

Some non-scientific calculators just use a left to right priority, so they will do the operations from left to right as they are written.  Avoid these left to right calculators like the plague!!

As below:

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Yes!  \(\frac{0.5}{0.5}=1\)

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As follows:

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Yes.  As long as you mean -(4!).

Factorials of negative numbers are not defined, so (-4)! isn't defined.

Ensure the calculator is in degree mode (its default) and enter M/60  where M is the number of minutes.

Here's a slightly different (but no quicker) way:

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Yes. Try  https://www.desmos.com/calculator   (though I didn't use this!)

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