Math Is Fun Forum

Similar to my last post (the question came in 3 parts), I worked this out, eventually, through trial and error, but I’m wondering what the method is.

My trial and error went something like;

9√3 suggested a fractional index
The √3 suggested 2 for the numberator of the fractional index
Then I had to try several values for the denominator before landing on 5, giving b=5/2
So, 3^5/2=9√3
Check; √3√3√3√3√3=√243
√243=√81√3
=9√3

I like the definition for 'angle' given here on Maths is Fun;

"The angle is the amount of turn between each arm," next to the illustration of said arms, joined at the vertex.

Another site defines the angle as THE SPACE between two intersecting lines.

Q. Do the lines (arms) in the MIFun illustration intersect? Or merely meet/join, at the vertex?

Another defines the angle as THE FIGURE FORMED when two rays share a common endpoint.

Q. Regards the two former definitions (amount of turn/space between); am I correct in thinking these don't include the arms and the vertex as part of the angle itself, the angle in the first defintion being how much the arms move, the angle in the second being how much space is created between the arms after they've moved?

Q. Regards the latter definition (the figure formed...); does this include the arms and the vertex and the space created/amount of turn, i.e, the whole thing?

Last Q. When a straight angle is shown in an illustration, i.e, basically a straight line, usually (always) with a semi circle added, and a dot in the middle of the line, should we see this as two arms opened up 'flat' with the dot as the vertex?

I know from my using my calculator that the answer to cube root 3 * cube root 3  is 2.08....

But I'm wondering if there's a method for multiplying two cube roots where the radicand is the same in each, something similar to the method for multiplying square roots where the radicand is the same, e,g;

sqrt 2 * sqrt 2 = sqrt 4 (which = 2)

or,

sqrt 12 * sqrt 12 = sqrt 144 (which = 12)

Is there such a method? Like;

cube root y * cube root y = cube root z ?

...the block at 5mph?

We’re told initially that velocity is speed, but with direction included in the measurement (that velocity is a vector quantity).

But then we’re told that velocity is the rate at which displacement occurs.

Which means if we run from our home to the park the velocity is >0 (so far so reasonable, for the novice student).

But if we run round the block, ending up back at our origin, the velocity is 0 (so far so confusing, for at least some novice students).

Q. When physicists agreed upon this (v=displacement/t) and it became standardized, did they do so having discovered a phenomenon about the world that was objectively true (v=displacement/t)? Or did they arbitrarily (not without good reason, of course, but arbitrarily in the sense that they could have framed it otherwise and it would still have worked) decide to frame things this way?

When an object in freefall, ignoring air resistance, accelerates from 0m/s to 1 m/s in its 1st second we are told that its average velocity is 5 m/s, and that it will have covered a distance of 5m. How is this calculated?

Make t the subject of the formula

p=(3-2t)/(4+t)

The left hand side was given in the traditional fraction form, with numerator on top of denominator; this is just the same, yeah, with the slash?

Here's my attempt;

Step 1
Mult.both sides by 4+t;
p(4+t)=3-2t
Step 2
Expand brackets;
4p+pt=3-2t
Step 3
Sub.4p from both sides;
pt=3-2t-4p
Step 4
Add 2t to both sides;
pt+2t=3-4p
Step 5
Factorise left side;
t(p+2)=3-4p
Step 6
Divide both sides by (p+2);
t=3-4p/p+2

I’ve come across questions recently that go something as follows;
Tom invests £2000 for 3 years.
For the first year he gets 3% compound interest...
Then for the next two years he gets x% compound interest. Etc.
I’ve been managing to get the correct answer, but the idea of someone receiving compound interest ‘for the first year’ seems strange to me.
Compound interest by it’s definition seems to me to only make sense if someone receives it for 2 or more years. If they are said to get it for ‘the first year,’ i.e, for 1 year, how does this differ from them getting simple interest for 1 year?
£2000. Compound interest. 3%. 1 year. Savings at the end of 1 year = £2060
£2000. Simple interest. 3%. 1 year. Savings at the end of 1 year = £2060

Is that correct?