Alan

625 = 5³*5 so its cube root is 5√5

135 = 3³*5 so its cube root is 3√5

Therefore  5 ³√625 + ³√135 = 5*5√5 + 3√5 = 25√5 + 3√5 = 28√5

x²+6x+2 factors nicely: (x+4)(x+2)

3x²+7x+2 also factors nicely:  (3x+1)(x+2)

The x+2 term is common to both, so it cancels from the original ratio which simplifies to (x+4)/(3x+1)

25^(4x-2) = 125^(3x-1)

(5²)^(4x-2) = (5³)^(3x-1)

5^2(4x-2) = 5^3(3x-1)

2(4x-2) = 3(3x-1)

8x - 4 = 9x - 3

-x = 1

x = -1

You could use the calculator on the home page here to solve this:

8255³³ ≈ 1.785*10¹²⁹

The exact number has 130 digits, which the calculator on the home page here will give you if you really want them!

You want the least common multiple (lcm) of 64, 84 and 96.  One way to do this is first to find the prime factors of each number:

64 = 2⁶

84 = 2²*3*7

96 = 2⁵*3

Now multiply together the largest powers of all the prime factors that occur:

2⁶*3*7 = 1344 this is the lcm

So they are all fired together again after 1344 seconds

You are probably thinking of the approximation that sin(θ) ≈ θ when θ is very small.  But this only holds when the angle is in radians.  For example suppose θ = 0.1 radians (=5.73°)

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{sin}}{\left({\mathtt{5.73}}^\circ\right)} = {\mathtt{0.099\: \!840\: \!745\: \!983}}$$ 

So sin(0.1radians) = sin(5.73°) ≈ 0.1

Or sin(0.1radians) ≈ 0.1

    sin(5.73°) ≠ 5.73

This is true whenever you have the sin of a small angle, no matter the context.

sin(60°) is +(√3)/2, so we want to look for the quadrants where sin is negative.  These are the 3rd and 4th quadrants.  

In these quadrants sin(180°+60°) = -(√3)/2 and sin(360°-60°) = -(√3)/2.

So we have that 7x = 240° and 7x = 300°

x = 240/7 ≈ 34.29° and x ≈ 42.86° 

Of course we can also add multiples of (360/7)° to these.

What you wrote was mm.  This means millimetres; metres would just be m (1mm = 0.001m).