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Put the fraction in brackets

Example

 1.           (3/4)^2

2.              (3/4)^(1/2)

N = 500r     where N is the total number of pages and r is the number of reams

The geometric mean of two numbers a and b can be interpreted as the length of a side of a square having the same area as a rectangle with a and b as its sides.

So the rectangle having 4 and 16 as its sides has an area of 64

And a square with area 64 will have a side =  to the square root of this, i.e., 8.

So.....8  is the geometric mean of 4 and 16

18  =  2 · 3 · 3

12  =  2 · 2 · 3

The lowest common multiple must have each factor the maximum number of times found in the problem.

18 requires at least 1 '2' and at least 2 '3's

12 requires at least 2 '2'x and at least 1 '3'

Taking the maximum number of each, the LCM needs 2 '2's and 2 '3's   --->   2 x 2 x 3 x 3  =  36.

3* 10= 30 and 3*1/2= 1 and 1/2. add them together for 31 1/2 is answer

To expand upon Anonymous's answer:

√28  =  √4√7  =  2√7 

(√28) · √7  =  (2√7) · √7  =  2 · (√7 · √7)  =  2 · 7  =  14

Anytime that you add a rational number (like 2/3, 14, 0, -7, etc.) to an irrational number (like √7, π, etc.), the answer will be irrational.

Potential Energy is measured in joules.

8 / (x - y)²  ÷ 12 / (x² - y²)

=  8 / (x - y)²  ·  (x² - y²) / 12

=  [ 8 · (x² - y²) ] / [ (x - y)²  · 12 ]

=  [ 8 · (x + y)(x - y) ] / [ (x - y)(x - y) · 12 ]

Reduce 8/12 = 2/3 and cancel the factors (x - y) from the numerator and denominator:

=  [ 2 · (x + y) ] / [ 3 · (x - y) ]

I assume what you have is this

[3(-0.1n^2+10n)]  * [-2(-0.2n+10)]     if so, we have

[-.3n^2 + 30n] * [-.4n -20]    let's just simplify this directly.....we have

.12n^3 - 12n^2 + 6n^2 - 600n =

.12n^3 - 6n^2 - 600n

And taking the derivative of this, we have

.36n^2 - 12n - 600

y  =  3(-0.1n² + 10n)   --->   y  =  -0.3n² + 30n   --->   y' =   -0.6n + 30

y  =  -2(-0.2n + 10)   --->   y  =  0.4n - 20   --->  y'  =  0.4

Total:  y'  =  -0.6n + 30  + 0.4   --->   y'  =  -0.6n + 30.4

If you add them first:  y  =  -.3n² + 30n + 0.4n + 10   --->   y  =  -.3n² + 30.4n + 10  

                                              --->   y'  =  -0.6n + 30.4

   ---   addd these together, don't multiply them ...

21²  =  (21x)² + (9x)²

441  =  441x² + 81x² 

441  =  522x² 

441/522  =  x² 

49/58  =  x² 

x  =  ±√(49/58)

Let the number of markers that Dave has = x  

So, Sheila has 5 times this number  = 5x

And we are told that

x + 5x  = 18

6x = 18      divide by 6 on both sides

x = 3

And Sheila has 5x  = 5(3)   = 15

Here's another - albeit, more exotic - way of determining the area of a rhombus.......

Given rhombus BCDE, let BK be the altitude of triangle BDE. And the area of BDE = (1/2)DE * BK. And by the same token, let OD be the altitude of triangle BCD. And the area of BCD = (1/2) BC *OD.  And OD = BK.......and let these be = h, the height of the rhombus. And DE = BC. So, the total area of the rhombus = (1/2)h (DE + DE) = h*(1/2)*2DE =  h * one side of the rhombus.

And we have triangle EHD with base ED and altitude HR. And HR is a radius of the circle with a center of H inscribed in the rhombus. And we have three more triangles, BHE, BHC and CHD with the same radius as an altitude and a side of the rhombus as a base. And since all the sides of the rhombus are the same, then the total area is just (1/2)(4DE)*r = 2DE * r = DE* 2r  =  the side of the rhombus * the diameter of a circle inscribed in the rhombus

This implies that

[h * one side of the rhombus]  = [one side of the rhombus * the diameter of a circle inscribed in the rhombus]

Which further implies that  the height of the rhombus = the diameter of a circle inscribed in the rhombus.

the answer to that problem is 21

8.75 is the answer

negative time a negative equals a positive. Or they cross out . equals 1

What are the dimensions of the shed? Do you want to include the area of the floor? Do you want to include the area of the roof? If so, what type of roof does it have?

The geometric of the numbers a and b is the number x that solves this equation:  a/x  =  x/b.

Another way to write this is:  x²  =  a·b  or  x =  ±√(a·b).

x  =  ±√(4·16)  =  ±√(64)  =  ±8

A bishop only threatens a piece of the same color; a knight only threatens a piece of the opposite color. So, the probability that they threaten each other is zero.

18

14 -(-4)

-(-) means to add so, 14 + 4= 18! :)

If this is the problem:

√(5x² - 7)  =  2x

Square both sides:

--->   5x² - 7  =  (2x)²

         5x² - 7  =  4x²

Subtract 5x² from both sides:

--->     -7  =  -x²

Divide both sides by -1:

--->   7  =  x²

Find the square root of both sides:

--->   x  =  ±√7

However, if you check by placing both √7 and -√7 into the original equation, only √7 works.