Alan

I've given my answer to your (almost certainly correct) interpretation also!  (11/72 or 0.153).

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If log_ab = x then a^x = b so if log_1/√20.125 = x then 1/2^x/2 =0.125  

Now 0.125 = 1/8 = 1/2³, so we must have x/2 = 3 or x = 6

Check

$${{log}}_{{\frac{{\mathtt{1}}}{{\sqrt{{\mathtt{2}}}}}}}{\left({\mathtt{0.125}}\right)} = {\mathtt{6.000\: \!000\: \!000\: \!000\: \!001}}$$

(The 1 at the end is just numerical error.)

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Let  x = 1   Then  x² = 1 as well

So x² - 1 = x - 1

The left-hand side can be written as (x+1)(x-1) so

(x + 1)(x - 1) = x-1

Divide both sides by x - 1 to get

x + 1 = 1

But we started by saying x = 1, so the left-hand side is 1 + 1 or 2.  This means the last equation is

2 = 1

This is the sort of thing that happens when you allow division by zero (which is what I did above when dividing through by x-1).

You could also solve this algebraically as follows.  Knowing that tan θ = sinθ/cosθ we can write:

cosθ - sinθ/cosθ = 0

Multiply through by cosθ

cos²θ - sinθ = 0

Also, cos²θ = 1 - sin²θ so

1 - sin²θ - sinθ = 0

Rearrange:

sin²θ + sinθ - 1 = 0

This is a quadratic in sinθ with solutions -(1-√5)/2 and -(1+√5)/2.  The second result is not a valid solution because it is outside the range ±1.

So taking the arcsine of the other result we get

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}^{\!\!\mathtt{-1}}{\left({\mathtt{\,-\,}}{\frac{\left({\mathtt{1}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{5}}}}\right)}{{\mathtt{2}}}}\right)} = {\mathtt{38.172\: \!707\: \!627\: \!012^{\circ}}}$$

There is another value at 180°-38.173° = 141.827°

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1/3 as a decimal fraction has an infinite number of recurring 3's.   1/3 = 0.33333..... the 3's go on forever!

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Just multiply 1840 by 1.2

Or calculate 0.2*1840 then add the result to 1840.

20% means 20/100 which is 0.2

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It's true as long as x is greater than zero.

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You are correct Mellie, though I recommend that in your first line that has numbers in you use brackets to ensure there is no confusion.

i.e. (-4-5)/(4--2)  is preferable to -4-5/4--2

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No, this question keeps cropping up.

(-3)*(-3) = +9

But

-3^2 means -1*(3^2)  = -9 for most calculators, because exponentiation (raising to a power) takes precedence over negation.

(Microsoft Excel spreadsheets provide a rare exception.  Excel gives a unary minus higher precedence than exponentiation!)

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If you were born in any year from 2000 to 2015 the probability of getting the 1st digit (2) is 1/10; that of getting the 2nd digit (0 ) is 1/10; that of getting the third digit (0 or 1) is 2/10, and that of getting the fourth digit (0, 1, 2, 3, 4, 5) is 6/10, so overall:

probability = 1*1*2*6/10^4 = 0.0012

Of course, if you are referring to someone like Chris or myself then the probability increases dramatically!

No! On second thoughts, the probability of matching your specific four digits is just 1/10 for each digit, so the probabilty is just 0.0001, whenever you were born!. (Chris and I are let off the hook!)

I must stop trying probability questions at night!  (Possibly at any time!)

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