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3.

Area   =  pi * r^2   ....so  we have

108.94  =  pi * r^2      divide both sides by pi

108.94 / pi  = r^2      take the square root of both sides

√ [ 108.94 / pi ]  =  r  ≈  5.89 ft 

4.  Note, NSS....we don't need to do much math here....logic will suffice

From  a  positive radical on the right, we CANNOT  get a negative answer....thus....the 5x  on the left side CANNOT evaluate to a negative whenever we plug in  an x value.....thus, x CANNOT be negative

Notice that the first, second and fourth answers will produce at least one negative outcome for 5x

So....by a process of elimination, the second answer is the only one that is any good

Do you see that???.....sometimes....logic is BETTER than MATH  !!!

5.  These are a little more difficult than what you have done before, NSS

First....we would like to separate the radical terms....so......add √x to both sides

√ [ 5x + 1 ]  =  5 + √x          square both sides

5x + 1  =  (5 + √x)^2

5x + 1  =   ( 5 + √x ) ( 5 + √x)   simplify

5x + 1  =  25  + 5√x + 5√x  +  x

5x + 1   =  25 + 10√x  +  x       

We want the radical on one side....so.....subtract 25, x from both sides

4x - 24  =  10√x       divide through by 2 to reduce the size of the numbers

2x -  12  = 5√x         

2 ( x - 6)  =  5√x          square both sides, again

2^2 (x - 6)^2  =  25x

4 (x - 6) ( x - 6)  =  25x

4 ( x^2 - 12x + 36)  =  25x

4x^2 - 48x + 144  = 25x      subtract 25 x from  both sides

4x^2 -  73x + 144  =  0    using the quadratic formula, the solutions to this are

x  = 9/4     or  x  =16

Since 9/4 isn't on the answer sheet......16 must be the only possible solution  !!!!!

Plug this into the original equation to make sure......!!!!

NSS.....I don't often recommend this....but...on a test.....because of the lengthy nature of this type of problem...I might be tempted to just plug the choices into the original equation to see which one was correct!!!!

1.  Notice, NSS,  that

4/7  * 4/7 * 4/7  * 4/7  =   256  / 2401

But also

-4/7 * -4/7 * -4/7 * -4/7  =    256 /  2407

So.....  (4/7, -4/7)  is correct

2.   √[ 36 g^6]   =  

Write  this as an exponential...it makes it easier to see, IMHO.....so we have

(36)^(1/2)  *  (g^6)^(1/2)  =

(6^2)^(1/2)  * ( g^3)  =

[ use   (a^m)^n   =  a^(m * n) ]

6 * g^3

OK, Manuel

1296  is even, so it's divisible by 2

648  *  2             factor 648 

324 * 2 *  2        factor 324 

162 * 2  * 2 * 2   factor162 

81 * 2 * 2 * 2 *2       factor 81

9 * 9  * 2 * 2 * 2 * 2    factor both 9's

3 * 3 * 3 * 3 * 2 * 2 * 2 * 2  =

3⁴ *  2⁴           [  3 and 2 are both prime.....no further integer factorization is possible ]

Now  if we wanted  √1296, we would have  √[ 3⁴ * 2⁴]  = 

√3^2  *  √3^2   *   √ 2^2  *    √2^2  =

√9 * √9 * √4 * √4  =

3 * 3 * * 2   *  2  =

36

Check for yourself that 36^2  =  1296

5.   9^1/3  *  81^1/3   =   ( 9 * 81 )^1/3   =   ( 9 * 9 * 9 )^1/3   =   (9³)^1/3   =   9^3/3   =   9¹   =   9

6.   $\frac{8}{\sqrt[7]{x^{15}}}\,=\,\frac{8}{x^{\frac{15}{7}}}\,=\,8x^{-\frac{15}{7}}$

Happy Birthday heureka!!!!

thank you <3

3.   Perimeter  =  2 ( W + L)

So  we have

2 ( √13 + 7√13 )

2 (8√13)  =

16√13  units

4.  √20 + √45  - √5  =   

{ The trick here, NSS, is to try to write  each radical in terms of the last one}

√ [ 4 *5 ]  + √ [ 9 * 5 ]   - √5  =

√4 *√5  + √9 *√5  -  √5   =

2√5  +  3√5  - √5  =

5√5  -  1√5   =

4√5

7.

P  =  d^3/2

When  d  =  6.76, we have

P  =  (6.76)^3/2    ≈  17.58

8.   Hard to read this...but I think we have

³√x³

____

⁵√x²

Writing  in exponential form, we have

x^(3/3)                                   a^m

_____              using            _____        =    a^(m - n)         we  have  

x^(2/5)                                   a^n 

x ^( 3/3  - 2/5)  =

x^(1 - 2/5)  =

x^(5/5 - 2/5)  =

x^(3/5)

9.  We want to multiply top/bottom by the congugate of 2 - √5 ⇒  2 + √5

2 + √5        (2 + √5)

_____

2 -  √5        (2 + √5)

Let's do these separately

( 2 + √5 )  ( 2   + √5)  =

2(2)  + 2√5 + 2√5  +  √5*√5  =

4 + 4√5 + 5 =

9 + 4√5        (1)

( 2 - √5 )  ( 2   + √5)  =

2(2)  - 2√5 + 2√5  -  √5*√5  =

4 - 5 =

-1      (2)

Put  (1)  over (2)  and we have

[ 9 + 4√5 ]                      -9  - 4√5

________       =     

       -1

7.    6.76^(3/2)  

10.

(8a^-3)^-2/3   =   8^-2/3  *   (a^-3)^-2/3

Note, NSS.....

a^-m   =   1 / a^m           and       ( a^m )ⁿ    =  a^{m * n}

So we have

1 / 8^2/3    *   a^6/3

And  we can write

1 / ³√ [ 8 ² ]     *  a²

1 / ³√64  * a²  

1 / 4  * a²

a² / 4

That's a really cool drawing! Why didn't you tell us you were an artist or post about it?

EP  is correct.....

Using the guest's "formula"....   24  would be    (1 - 1) * 100%  =   0%  of itself !!!!

Happy Birthday, heureka.....!!!!!!

Second one :

Notice that if we let the normal sine graph have an amplitude of 1/2 and then  "flip" - reflect - it across the x axis, we get the "negative" sine graph with an amplitude of 1/2

So....the correct answer is

f(x)  = ( -1/2 ) sin (x)

72/24 x 100% = 300% of 24

= 16k^2 - 8k +1 = 4k^2

12k^2 -8k +1 =0      Use quadratic formula to find

k = 1/2  or  1/6

Ya got the first one.....but not the second one......  wanna try again?

The September price was 'x'     the October price was   1.2 * x

1.2 * x = 1.50

x = 1.50/1.2 = $1.25   (sept price)

September earnings     1.25 * 40% x 900 = $450.

October earnings          1.50 * 40% x 700 = $420        They earned $30 MORE in Sept

72 / 24 = 3 - 1 x 100 = 200%

same he is my fav out of them

We need the point(s) where   y = 5x+6    equals  y = 9x-32    

     or    5x+6 = 9x-32       solve for x     add 32 to each side of the equation

            5x+38 = 9x           subtract 5x from both sides

                  38 = 4 x

                    9.5 = x           Now substitute theis 'x' into one of the equations to find 'y'

          y= 5(9.5) +6        y= 53.5      so   x,y  =  9.5 , 53.5   

  (you can sub 'x' in to the other equation to see if you get the same result as a check)

0.5x+1.5= -4(x-3)

                = -4x +12       add 4x to both sides

4.5x +1.5 =12                 Subtract 1.5 from both sides

4.5x = 10.5                      Divide by 4.5

x = 2 1/3

then sub  this 'x' in to one of the other equations to find 'y'

y=-4(x-3)=-4x+12

 = -4 (2 1/3) + 12 = 2 2/3       x,y = 2 1/3 , 2 2/3           

5w^-2 y^6 * 2w^5 y

$5w^{-2} y^6 \times 2w^5 y\\5w^{-2} y^6 \times 2w^5 y^1\\ =5\times 2w^{-2+5} y^{6+1}\\ =10w^{3} y^{7}\\$