a rectangle has a length of x3y4 inches a width of xy7 inches. which expression represents the ratio of the length of the rectangle to the width of the rectangle?

R = x^2/y^3

Step-by-step explanation:

here we will use the relation:

x^a/x^b = x^(a - b)

We know that:

Length = x^3*y^4

Width = x*y^7

And we want to find an expression that represents the ratio between the length and the width.

Then we just need to take the quotient Lenght/width or:

R = ( x^3*y^4)/(x*y^7)

Now first let's distribute in such way the x's and the y's are separated, and then let's use the relationship that is above.

= (x^3/x)*(y^4/y^7) = x^(3 - 1)*y^(4 - 7) = x^2*y^-3

R = x^2/y^3

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

if x = 1, then the answer is 3.5