All Answers 358384 Answers

43,560 sq ft

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A little experimentation shows that the successive powers of 490497 end in 1, 7, 9, 3, 1, 7, 9, 3, 1, 7, 9, 3, .... etc.

So if the remainder of the power, when divided by 4 is zero the result ends in 1, if the remainder is 1 the result ends in 7, etc.

Now, mod(252,4) = 0, so the units digit of 490497²⁵² is 1

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Here's a 3d plot of the function (in blue) on the left-hand side of the equation, showing where it intersects with the zero plane (in grey).  There is also a 2D plot showing the lines of theta vs beta on the zero plane.  The axes are in radians (0 to pi/2 for both theta and beta).

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good to know.

x/8+x/10=9/8    multiply through by the common denominator of 8 and 10   = 40

40(x/8) + 40(x/10) = 40(9/8)

5x + 4x  = 45       sinplify

9x = 45       divide both sides by 9

x = 5

23 / 27  = about 85.2%

√(25/81)  = √25 / √81  =    5/9

4.28710198105953E+62

cos2x+5cosx+3=0   ....note that cos2x = 2cos^x - 1  ....so we have.......

2cos^x - 1 + 5 cosx + 3  = 0     simplify

2cos^x + 5cosx + 2   = 0      factor

(2cosx + 1) (cosx + 2)  = 0     setting each factor to 0, we have......

2cosx + 1 = 0                             and                           cosx + 2 = 0

subtract 1 from each side                                            subtract 2 from each side

2cosx = -1                                                                cos x = -2

divide both sides by 2                                                this has no solution

cosx = -1/2

And the solution to this is  x  = (2/3)pi + 2pi * n    and     x = (4/3)pi + 2pi * n ....   where n is any integer

$${\frac{{\mathtt{14}}}{{\mathtt{84}}}} = {\frac{{\mathtt{1}}}{{\mathtt{6}}}} = {\mathtt{0.166\: \!666\: \!666\: \!666\: \!666\: \!7}}$$

$${\sqrt{{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{11}} = {\mathtt{0}}$$

$${\sqrt{{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}}} = {\mathtt{11}}{\mathtt{\,-\,}}{\mathtt{x}}$$           |  quadrieren

x + 1 = 121 -22x +x²

x² -23x - 122 = 0

x(1) = 23/2 + sqrt( 11.5² +122 )             $${\frac{{\mathtt{23}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\sqrt{{{\mathtt{11.5}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{122}}}} = {\mathtt{27.445\: \!218\: \!719\: \!101\: \!974\: \!1}}$$

x(2) = $${\frac{{\mathtt{23}}}{{\mathtt{2}}}}{\mathtt{\,-\,}}{\sqrt{{\left({\frac{{\mathtt{23}}}{{\mathtt{2}}}}\right)}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{122}}}} = -{\mathtt{4.445\: \!218\: \!719\: \!101\: \!974\: \!1}}$$

the answer is 1.18 

Simple one $${\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{3}} = {\mathtt{\,-\,}}\left({\mathtt{2}}{\mathtt{\,\times\,}}\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}\right)\right) \Rightarrow {\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{7}}}{{\mathtt{3}}}} \Rightarrow {\mathtt{x}} = -{\mathtt{2.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

x=-2.3 and the other x is-7/3

Angelo

$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{3}}} = {\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{{\mathtt{3}}}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\mathtt{0}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{1.5}}\\
{\mathtt{x}} = {\mathtt{0}}\\
\end{array} \right\}$$

Normal and cool

Rosala, why does TR need to eat almonds?

Area _trapezoid =  [height/2] * [ sum of the bases]  .....so we have.....

[5/2] * [7 + 6]  =

(5/2]*[ 13]  =

65/2  sq ft = 32.5 sq ft

(7 + 2/3) /  5 =

(23/3) / (5/1)  =  [ remember, that when we divide fractions, we "keep the first,"  "change to multiplication" and "flip" the second fraction ]......so, we have.......

23/3   * (1/5)   =

23/15 

log₃ 4    .....we can use the "change-of-base" rule here    ......we have.....

log 4 / log 3   =  about  1.26186

10m^2 - 12m - 12 = 0      divide every term by 2

This will not factor.......using the onsite solver and the quadratic formula, we have these two solutions :

$${\mathtt{5}}\left[{{m}}^{{\mathtt{2}}}\right]{\mathtt{\,-\,}}{\mathtt{6}}{m}{\mathtt{\,-\,}}{\mathtt{6}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{m}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{39}}}}{\mathtt{\,-\,}}{\mathtt{3}}\right)}{{\mathtt{5}}}}\\
{\mathtt{m}} = {\frac{\left({\sqrt{{\mathtt{39}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{m}} = -{\mathtt{0.648\: \!999\: \!599\: \!679\: \!679\: \!6}}\\
{\mathtt{m}} = {\mathtt{1.848\: \!999\: \!599\: \!679\: \!679\: \!6}}\\
\end{array} \right\}$$    

2x^3=3x^2    rewrite   as

2x^3  - 3x^2  = 0     factor

x^2 (2x - 3)  = 0       set both factors to 0.....and we have.....

x^2  = 0                                              and                            2x - 3 = 0

take the square root of both sides                                          add 3 to both sides

x = 0                                                                                    2x = 3

                                                                                          divide both sides by 2

                                                                                             x = 3/2

So.....the two solutions are   x = 0 aand x = 3/2

dividend / divsior   = quotient      ...... so........

dividend = quotient * divisor

Example :

dividend / 4 = 8     .....so......

dividend = 8 * 4    =  32      and this is the dividend !!!

BTW...I love "Ineeda_sandwich"     .....great screen name...  Welcome to the forum......!!!

x-3=-2(x+5)    simplify

x - 3 = -2x - 10      add 3, 2x to both sides

3x  = -7      divide both sides by 3

x = -7/3

You need to eat almonds tin.....he's still in my shield :P

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