CPhill

125x^6   =  125x⁶

As Severoth says......this is a representation of an exponent

1. 

SAS Similarity Theorem

2.   

40/ 32  =  x / 4       multiply both sides by   4

      4 * 40  / 32  = x

      160 / 32 = x

        5  =  x

3.  Triangle ABC  is similar to triangle DFG

So...... each side of DEF  is 6 times that of the corresponding side of ABC

So  DG  =  6AC  =  6*6   =  36

So.....the perimeter of  DFG  =  18 + 24 +  36   =  78

4.  A segment drawn parallel to a base splits the sides in  corresponding ratios

 AB / BD  =  AC / CE      .....so......

AB / 15  = 8 / 10       multiply both sides by   15

AB  = 15 * 8  / 10    =   120/10  =   12

5.  A bisected apex angle sets up the following ratio

AC / BC  = AD / BD

5 / 6  = x / 4.2       mltiply both sides by  4.2

4.2 * 5  /  6  = x    

21 / 6  =  x  =  7/2  =   3.5

6.  

8 / 12  =  [ x + 4 ] / [ 2x + 1 ]     cross-multiply

8 [ 2x + 1 ]  = 12 [ x + 4 ]          simplify

16x  +  8   =  12x   +  48        subtract 8, 12x   from both sides

4x  =  40          divide both sides by   4

x  = 10

7.   path length   =  √ [ 65^2 + 34^2  ]  =  √ [ 5381  =  73.4 m

What's the question  ???

[ x √y ]  / [ y √ x ]

Write in exponential form

[ x * y^1/2 ] / [ y * x^1/2 ]       and we can write

[ x^1 / x^1/2 ]  *  [ y^1/2  / y^1  ]

Remember that        a^m / a^n   =   a^( m - n)

So we have

x^(1 - 1/2)  *  y^(1/2  - 1)

x^1/2   *   y^(-1/2) =

√x / √y          multiply top/bottom by √y

[ √ x√ y ]  / y

√[ xy] / y  ⇒   "C"

Nice, killersteve....!!!

First one

2. Corresponding Angles

3.  Triangle JKL  similar to Triangle JMN            AA similarity

4.  Corresponding Parts of Similar Triangles form equal ratios

5. Segment Addition  Postulate

6. Substitution

9. Subtracion Property of Equality

Second one

They are similar by AA  congruency

The areas  are in ratio     PQ^2 / AB^2  =    (4k)^2 / 4^2  =  16k^2 / 16  =  k^2

Probably just a coincidence......

(5x^2  - ab)^3     

Let  a  =  5x^2         and  b  =   (ab)  

So     (a  - b)^3   =   a^3  -  3a^2b   +  3ab^2 -  b^3

So we have

1 * (5x^2)^3  -   3(5x^2)^2*ab  +  3(5x^2)*(ab)^2 -  1(ab)^3  =

125x^6  -  75x^4ab  +  15x^2(ab)^2  - (ab)^3

You're welcome, OBT...!!!

a4 / a2  =  3

Find

a5 / a3

a4 / a2   = 3   ⇒    a4  =  3a2   (1)

a2  +  2d   = a4      sub (1)  into the right side

a2  + 2d  =  3a2

2d  =   2a2

d  =  a2

So

a5  =  a2 + 3d   =   a2 + 3a2  =  4a2

a3  =  a2 + d    =   a2  + a2  =   2a2

So

a5 / a3  =   4a2 / 2a2   =  2