We can write some natural numbers as the difference of two perfect squares in this way. Eg: Take the number 33

33 = 33×1

Let x+y=33 and x-y =1

we get,

x= 33+1/2 = 17 and y= 33-1/2 =16

{33 = 17²-16²}

If we choose

33 = 11×3

Let us take x+y=11 and x-y =3

x=11+3\2 = 7 and y= 11-3/2 =4

33=7²-4²

Using this, express the following numbers as the difference of two perfect squares.

a. 27
b. 47​

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

6 meters

step-by-step explanation:

the answer is; biodiversity resulting from few ancestors

adaptive radiation occurs when a population is subjected to a large natural environment with spatial ecological variations. the parts of the population in the different ecologies begin to adapt to their local environments. due to non-random mating across the large populations, this causes the different populations from the large population to speciate within their local environments. they, therefore, diverge to different species sharing a common ancestor.  

question 1: step-by-step explanation:

1/4 of 84 = 84÷4= 21