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Explanation for the answer:

\(\sqrt{64\times 81}\\ =\sqrt{64}\times \sqrt{81}\text{(Property of square root)}\\ = 8 \times 9\\ = 72.\)

\(\dfrac{2}{3} \div \dfrac{1}{2}\\ = \dfrac{2}{3} \times\dfrac{2}{1}\\ =\dfrac{2\times 2}{3}\\ =\dfrac{4}{3}\\ \text{OR}\\ =1\dfrac{1}{3}\)

n + 8 is the most simplified form.

When n = 1

n + 8 = 1 + 8 = 9

When n = 2

n + 8 = 2 + 8 = 10.

Like this.

When n = 10000000000,  n + 8 = 10000000008 LOL

This equation is linear.

The graph

= Draw a straight horizontal line passing through (0, -1)

\(\dfrac{0.6^{\frac{1}{2}}}{0.6^{\frac{3}{2}}}\\ = 0.6^{\frac{1}{2}-\frac{3}{2}}\\ =0.6^{-1}\\ = \dfrac{1}{0.6}\\ = \dfrac{5}{3}\\ \text{OR}\\ = 1\dfrac{2}{3}\)

1/2 x 8 x 9 x 10

= 4 x 9 x 10

= 36 x 10

= 360.

I mean multiply by 'x'

I see this is a sequences question but the right hand side have no 'n'!!

Gradient of line L......

You mean slope of line L?

3y = 5x - 6

y = (5/3) x - 2

This is linear and have slope-intercept form. Linear functions in slope intercept form have the form y = mx + b where m is the slope(gradient) and b is the y intercept.

Therefore gradient of line L is 5/3.

it should be 'in terms of x and y', nice assumption :)

Hi Alan .  I didn't see your answer.

My interent is sooooooooooooooo  slow tonight.

I am going to die of old age before it does .... anything      

5ysquared+10y+4=0 what is the answer

\(5y^2+10y+4=0\\ y=\frac{-10\pm \sqrt{100-80}}{10}\\ y=\frac{-10\pm \sqrt{20}}{10}\\ y=\frac{-10\pm2 \sqrt{5}}{10}\\ y=\frac{-5\pm \sqrt{5}}{5}\\ \)

2,000 - 100 =1,900

1,900 / 9 =211 1/9 drop the 1/9 fraction. That leaves 211 multiples of 9

The first one =108

The last one =1,998

[1,998 - 108] /9 =210 +1 =211 You have to count both the 1st and the last. Using the formula above:

1,998=108 + 9(n-1), solve for n:

n=211

Lets look at it a different way.

k = -4 |k - 2| x |3k| -1

\(k = -4 |k - 2| x* |3k| -1\)

3k and  (k-2)  are both positve  when  k>2

3k and  (k-2)  are both negative  when  k<0

so for    0<k<2  the equation becomes

\(k = -4 (k - 2) (3k) -1\\ k = -12k (k - 2) -1\\ k = -12k^2 +24k -1\\ 0= -12k^2 +23k -1\\ \text{Consider the parabola}\\ y= -12k^2 +23k -1\\ \)

This is a concave down parabola

axis of sym     y= -23/-24  =  23/24    this is between 0 and 2

when k=0   y=-1 

when k=2  y = -48+46-1<0

So no solutions here.

Between k=0 and k=2

3k >=0 and  (k-2) <=0  

so the equation becomes

\(k = 4 (k - 2) (3k) -1\\ k = 12k (k - 2) -1\\ k = 12k^2 -24k -1\\ 0= 12k^2 -25k -1\\ \text{Consider the concave up parabola}\\ y= 12k^2 -25k -1\\ \text{axis of symmetry=25/24 and this is between 0 and 2}\\ \text{When k=0 y=-1, when k=2 y=48-50-1=-3}\\ \text{this means that between k= 0 and k=2 the y value is always negative }\\\)

No solutions here either.

Here is the graph

16^(3/4) → (16^(1/4))^3 → 2^3 → 8

0.04^(-1/2) → 1/0.04^(1/2) → 1/0.2 → 5

I'll leave you to multiply these together.

.

No real solutions:

Using Pythagoras we have: 

x^2 + (3x)^2 = 25^2

or:    10x^2 = 625

            x = sqrt(62.5) → 7.906

The perimeter is now found from 4x + 25

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Just type solve(5*y^2+10*y+4) into the calculator on the home page here and then press enter.

.

 answer is 2

solution-  since sqrt(a)  / sqrt(b)  = sqrt (a/b)

then k = sqrt(40/10) = sqrt (4)  =2

k = -4 |k - 2| x |3k| -1

First I will do it the super long way.

I am going to split this into bits

k>=2

0<x<2

k<=0

\(If \;k\ge2 \;then\\ k = -4 |k - 2| x *|3k| -1\\ k=-4(k-2)*3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ k=\frac{-23\pm \sqrt{23^2-4*-12*-1}}{2*-12}\\ k=\frac{-23\pm \sqrt{528-48}}{-24}\\ k=\frac{-23\pm \sqrt{480}}{-24}\\ k=\frac{-23\pm 4\sqrt{30}}{-24}\\\)

\(k\approx\frac{-23\pm 21.9}{-24}\\ \text{neither of these answers are greater or equal to 2 so no solution here}\)

If  0<x<2

\(k = -4 |k - 2| * |3k| -1\\ k = -4*-(k - 2) * 3k -1\\ k = 12k(k - 2) -1\\ k = 12k^2-24k -1\\ 0 = 12k^2-25k -1\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k=\frac{25\pm\sqrt{625+48}}{24}\\ k \approx \frac{25\pm25.9}{24}\\ \text{Neither of these answers are between 0 and 2. No solutions here either.}\)

\(If \;\; k\le0 \;\;then\\ k=-4|k-2|*|3k|-1\\ k=4(k-2)*-3k-1\\ k=-12k(k-2)-1\\ k=-12k^2+24k-1\\ 0=-12k^2+23k-1\\ 0=12k^2-23k+1\\ k=\frac{23\pm\sqrt{529-48}}{24}\\ k=\frac{23\pm\sqrt{481}}{24}\\ k\approx\frac{23\pm21.9}{24}\\ \mbox{No solution here either }\)​

No solutions      

First find the mean:   mean = (10.56 + 9.52 + 9.73 + 9.80 + 9.78 + 10.91)/6 → 10.05

Now find the absolute value of the deviations of each value from the mean:

|10.56 - 10.05| = 0.51

|9.52 - 10.05| = 0.53

|9.73 - 10.05| = 0.32

|9.80 - 10.05| = 0.25

|9.78 - 10.05| = 0.27

|10.91 - 10.05| = 0.86

The maximum absolute deviation is 0.86, so d = 0.86

.

What is (5x)^-2?

\((5x)^{-2}=\frac{1*(5x)^{-2}}{1}=\frac{1}{1*(5x)^{+2}}=\frac{1}{25x^2}\)

When I was bored in science class I played around with a graphing calculator, and I found an interesting way to make line segments: \(y={sin}^{-1}(x)^0\)

The output was a line similar to y=1, but with a domain of -1<x<1(excluding 0). I later tried this on desmos.com, but it didn't work. However, for some reason, you can get the exact same effect with: \(y={{sin}^{-1}(x)\over {sin}^{-1}(x)}\)even though it is practically the same thing! This makes me wonder if this is actually a line segment function or if it is just a bug in how graphing calculators work.

It is good that you are playing with maths - that will help you learn.  You should join up here and become known to us. :)

Here are the 2 graphs that you talk about.

inverse sine can oly go from -1 to +1

If you think just of the right angled triangle.  

sin (angle)=opp/hyp

the hypotenuse has to be longer than the opposite side so this ratio must be les than 1

The use of sine is extanded past this but still the ration must be between -1 and 1

No look at the next bit

The top is the same as the bottom.  When you divide a number by itself the answer is 1 EXCEPT if the number is 0.

You cannot divide by 0!   So there is a hole in the graph where sin^-1x=0  and that is where x=0

I made the hole more obvious - desmos does not show it very well except if you click the point on the graph when you are actually in Desmos it will tshow you the hole :)

The calculator is on the home page :)