hectictar

What about this? I'm not sure if it counts as an algebraic rule!

$y=\max\left(-\left|2x-3\right|+2,1\right)$    on the unbroken domain  (0, 2)

perimeter of a square with side  s  =  4s

perimeter of a square with side  x + 3   =   4(x + 3)   =   4x + 12

10 + 2 × (3 ^ 2 - 2)

=    10 + 2 × (9 - 2)

=    10 + 2 × 7

=    10 + 14

=    24

Anytime you forget exponent rules, try writing the whole thing out like this:

=   (2a³)⁴

=   (2a³)(2a³)(2a³)(2a³)

=   (2aaa)(2aaa)(2aaa)(2aaa)

=   (2 * 2 * 2 * 2) * (aaa * aaa * aaa * aaa)

=   2⁴ * a¹²

=   16a¹²

And here's the shortcut method:

=   (2a³)⁴

=   2⁴ a^{(3 * 4)}

=   2⁴ a¹²

=   16a¹²

We want to find  a  such that the numerator is also zero when  x = 3

That is, we want to find  a  such that...

2(3)² - 3a(3) + 3 - a - 1  =  0

18 - 9a + 3 - a - 1  =  0

20 - 10a  =  0

20  =  10a

2  =  a

So yes, when  a = 2,  the limit exists.

Check:

$\lim\limits_{x\to3}\frac{2x^2-6x+x-3}{x^2-2x-3}\ =\ \lim\limits_{x\to3}\frac{2x(x-3)+1(x-3)}{(x-3)(x+1)}\ =\ \lim\limits_{x\to3}\frac{2x+1}{x+1}\ =\frac74$

If you enter  10000  into the calculator on this site and press enter, it will give you  1e4  (which is equal to  10e3)

speed  =  distance / time

acceleration  =  [ speed ] / time  =  [ distance / time ] / time  =  distance / time²

force  =  mass * [ acceleration ]  =  mass * [ distance / time² ]  =  mass * distance / time²

work  =  [ force]  * distance  =  [ mass * distance / time² ] * distance   =   mass * distance² / time²

power  =  [ work ] / time  =  [ mass * distance² / time² ] / time   =   mass * distance² / time³

Mass is measured in kilograms.

Distance is measured in meters.

Time is measured in seconds.

power  =  mass * distance² / time³   =   mass * distance² * time^-3   =   kg * m² * s^-3

power   =   kg * m² * s^-3

It is a 42-inch TV. This means the length of the diagonal is 42 inches.

Aspect ratio is the ratio of the width to the height.

So if we let the unknown scale factor be called  s ,

then  the width  =  4s

and  the height  =  3s

Now we can use the Pythagorean Theorem to find what  s  is.

(4s)² + (3s)²  =  42²    This is the equation given in the hint.

(4s)(4s) + (3s)(3s)  =  42²

16s² + 9s²  =  42²

                                   Just like  16 apples + 9 apples = 25 apples, so does  16s² + 9s² = 25s²

25s²  =  42²

                       Let's rewrite  25s²  like this...

(5s)²  =  42²

                       and take the square root of both sides.

5s  =  42

s  =  8.4    (inches)

Now that we know what  s  is, we can find the width and the height.

width  =  4s  =  4(8.4)  =  33.6    (inches)

height  =  3s  =  3(8.4)  =  25.2    (inches)

Here's a pic:

If  PQ = 12 ,  then the area of              is $\mathcal A$

Is something supposed to go in the blank?

Euclid gave us a definition of a "rectangular parallelogram" – a "right-angled parallelogram" which we now call a "rectangle" – that does not exclude squares, and this was written around 300 BC (much longer than 55 years ago!).