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2 + sqrt 2   +  1/(2 + sqrt 3) * ( 2-sqrt 3)/(2 -sqrt 3)  +  1/ (sqrt 5 - 2) * (sqrt 5 + 2)/(sqrt 5 +2)  =

2 + sqrt 2  + 2 - sqrt 3  +  sqrt 5 + 2   =

6 + sqrt 2 - sqrt 3 + sqrt 5

2^(16x) = 128 * 4^x

2^(16x)  2^7 * (2^2)^x

2^(16x)  = 2^7 * 2^(2x)

2^(16x)  = 2^(7 + 2x)

16x  = 7 + 2x

14x  = 7

x = 7/14 =  1/2

2^x  = 2^(1/2)  =  sqrt 2

The function f(x)f of x 𝑓(𝑥) is increasing and invertible. 

The function is even, increa sing on [-10,0]open bracket negative 10 comma 0 close bracket, decreasing on, and not invertible.

a + b  = 14

b = 14  - a

a^3  + (14 -a)^3  = 812 + a^2 + (14 -a)^2

a^3  - a^3 + 42a^2 -588a + 2744  = 812 + a^2 + a^2 - 28a + 196

40a^2  - 560a + 1736  =  0 

5a^2  -  70a + 217  =  0

a  =  [  70  +  sqrt  [  70^2  - 20*217] ] / 10   =  [ 70 + sqrt [ 15260] ] / 10  =  [ 70 + 2sqrt 35 ] / 10  =

7 +  sqrt (35) / 5

And  b =  7 - sqrt(35) / 5

Also   a =  7 - sqrt (35) / 5     and b = 7 + sqrt (35)   / 5

The answer is 19/3.

Express \log_{15} 72 in terms of P and Q.

\(P = \log_8 3\\ 8^{log_8\ 3}=3\\ 8^P=3\\ \color{blue}8=3^{\frac{1}{P}}\)

\(Q = \log_3 5\\ \color{blue}3=5^{\frac{1}{Q}}\)

\(x=\log_{15} 72\\ \color{blue}15=72^{\frac{1}{x}}\)

will be continued

I can't go any further. Sorry!

 !

Find the radius of the semicircle.

Let the side length of the squares be s.

Case A) Squares next to each other.

Then the radius of the semicircle is \(\color{blue}r=s\sqrt{2}.\)

Case B) Squares on top of each other.

Then the radius of the semicircle is:

\(r^2=(2s)^2+(\frac{s}{2})^2\\ r=\sqrt{\frac{4\cdot4s^2+s^2}{4}}\\ \color{blue}r=\frac{s}{2}\sqrt{5}\)

 !

Couldn't you just substitute a=-14-b into the second equation and then factor/solve the cubic?

(A) k > 0 and a > 1

(B) k < 0 and a < 1

(C) k < 0 and a > 1

(D) k > 0 and a < 1

(x^4 + x^3 + x^2 + x + 1) + (x^4 - x^3 + x^2 - x + 1)

\(2x^4 + 2x^2 + 1\)

So far I get \(a_1 = 5\), \(a_2 = 10\), and \(a_3 = 20\), getting \(35\). Not sure.

\(1.11111 + 0.11111 + 0.01111 + 0.00111 + 0.00011 + 0.00001 = 1.23456\)

L0  =  2

L1  =  1

L2  =  3

L3  =  4

L4  =  7

P = 2^(1/2) * 4^(1/4) * 8^(1/8)* 16^(1/16)

P =  2^(1/2) * (2^2)^(1/4) * (2^3)^(1/8) * (2^4)^(1/16)

P = 2^(1/2)* 2^(1/2) *2^(3/8) * 2^(1/4)

P = 2^[ ( 4 + 4 + 3 + 2)  / 8 ]

P = 2^( 13 / 8)

P =   ⁸ √ (2^13)

a + b =    8  + 2^13   =  8 + 8192  =   8200

\(P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81} \)

\( \sqrt[a]{b} \)

P  =  3^(1/3) *  (3^2)^(1/9) * (3^3)^(1/27) * (3^4)^(1/81)

P = 3^(1/3) * 3^(2/9) * 3^(1/9) *3^(4/81)

P = 3 ^( 1/3 + 2/9 + 1/9 + 4/81)

P = 3^ [ ( 27 + 18 + 9 + 4 ) / 81 ]

P = 3 ^ [ 58 / 81]  =   ⁸¹ √ (3*58)

a + b =   81 + 3^58  =   81 +   4710128697246244834921603689    =  4710128697246244834921603770

a1  =  1

a2 =   2

a3  = a2 * (2/1)   =  2 * 2 =  4

a4 =   a3 + (a3 -a2) =  4 + 2 = 6

a5 =  6 * (6/4)  =  9

a6  = a5+ (a5 -a4)  = a5 + 3  = 12

The series is

a1   a2   a3   a4    a5    a6  ......

1     2     4      6      9     12

The answer is 884.