Math Is Fun Forum

Please do not use a period for attempted multiplication.  What you indicated to the right of the equals sign is unambiguously
equivalent to the use of decimal numbers for the bases as in "seven point seven raised to the k exponent minus two point two
raised to the k exponent equal to ..." and so on.

Please use multiplication dots as Monox D. I-Fly did instead.  Or, use an asterisk or bracketing such as parentheses.

Here are a couple of suggested fixes:

Notice in the "behind-the=scenes" text how the use of inserting the back slash tightens the asterisks between characters, while at the same time spreading
the other characters out for better readability.

iamaditya, it has been four and a half months since your last reply to bobbym, who has since died.

The answer to # 15 is log(12).

These are incorrect.  All of the forms involving the natural logarithm for the answer must have absolute value
bars where the logarithm of the quantity is being taken of in these examples, not just parentheses.

This goes for the rest of your posts in this thread.

If these two lines are in the same plane, and they are not the same line, what can you state about them?

You cannot see the circumference of the fill circle/disk even though it
is not covered. Again,
the semi-disks do not include their circumferences.
And you cannot see the diameter between the two semi-disks.
It is also not covered.  Not being able to see with our eyes is not proof.

[1 - cos(x)]÷[sin(x)]^2

Your next post to the OP after the one above makes no sense.  You can't just take the word of a user whose own interest is to get answers on something
without disclosing proof of A) where these problems are really from and B) whether there is documented permission from a third source to use the problems.

The OP's reply "They do not mind" gives you zero assurance, and you could still be in trouble for discussing the answers.  Why would you entertain trusting
a user's word on likely sensitive material!?

You must have an error somewhere.  I know I am asking about all integers, except zero.

But it should not make a difference if I were to ask about a, b being negative, c is positive and all three belonging
to the set of real numbers.  And, I will make sure [in general, and in my example below] that a < b < c.

I came up with this counterexample for you:

If you work this out, you will see that the expression on the left-hand side simplifies to 1.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

In general where a and b negative and a < b:

Let r, p be positive integers

Let b = -r

To make sure a < b,

let a = -p - r

c = c

Substitute:

The sides do not contradict each other in signs.

If I were to multiply each side by cp(p + r), I would get:

You're not going to manipulate that and get a negative LHS versus a positive RHS.

If I want a, b, c to be positive integers for other problems, I will state "positive integers" in those cases.
It seems some younger students aren't aware of negative integers, and I could see the clarification question
being asked here because of that.

Here, I included all integers, except zero.

So, it has been established that positive integers don't work, had that been the question.  (I had already
researched that angle of the question some days prior.)