Thanks Alan.  .

I'd like someone to point out the flaw in fiora's working. I agree there exist more than one answer, but where is the error in fiora's calculation if the result obtained there is not one of the valid answers?

Wolfram alpha doesn't agree with any of us.   

FWIW, I got the same answer as fiora.

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Determine x such that  10^x = 2.278768 x 10⁴²

So, taking log_x of both sides, we have

x·log₁₀(10) = log₁₀(2.278768) + 42•log₁₀(10)

x = 0.3577001 + 42

   = 42.3577001   (Answer)

It's difficult to check this on a calculator, so to verify the approach repeat it but using an example you can realistically evaluate on a calculator, e.g.,

Determine x such that  10^x = 2.278768 x 10⁴

and if after you determine the value of x you check that the right side equals the left here, you may conclude that your answer in the first part should be right, too.

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Suppose you were required to add:

7 + 6 + 24 + 12 + 1

You could regroup as:

(7 + 12 + 1) + (6 + 24)

This is now easy to add in your head:

(20) + (30)

= 50  (Answer)

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There is a button that looks like ^y√x so use that to find the y^th root of x.

But for integer roots there is an alternative way using the x^y button beside it.

Example: There are two ways to write the 4^th root:

  ∜57 ≡ (57)^¼

So after taking the reciprocal of y you can make use of that x^y button.

Does this answer your question? 

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Good point, Melody.

We would need to involve the quadrant diagram for a complete answer.

You could use a rule for this, but it is more instructive to work it out from first principles:

draw a right-angled triangle, choose one of the acute angles to be our angle x, so denote the length of its ADJACENT side as 0.6372, and the length of the HYPOTENUSE as 1.000 . This means we've made cos x = 0.6372

Use Pythagoras' Theorem to determine the length of the unknown side.

Inspect the triangle and write down the ratio of the two sides that determine sin x,

i.e., opposite/hypotenuse

Done!

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Melody, the original thread was back here:

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