Alan

Let A = number of Ava's books and E = number of Emily's books:

(2/7)*A = (1/9)*E

A = (7/2)*(1/9)*E

A/E = (7/2)*(1/9)

A/E = 7/18  or A:E = 7:18

(Note that Emily must have more books than Ava). 

Use Ohms law:  V = I*R

1. Find R:   R = V/I  → 24/0.008  ohms

2. Find new voltage:  V = I*R → 0.013*24/0.008 or V = 39 volts

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You are right GingerAle - these multiple exponentiations are making my brain swirl!

Just type in round(30/11) and press =.    It works!

Let V be full volume.

We are told:  V/2 -420 = V/8

Subtract V/8 from both sides: V/2 - V/8 - 420 = 0

Add 420 to both sides:  V/2 - V/8 = 420

Simplify the left hand side:  3V/8 = 420

Multiply both sides by 8/3:  V = 420*8/3

Simplify: V = 140*8  or  V = 1120 gallons

In Guest #1 #2 calculator he/she put the terms as (4²)^{log₂log₂7} + (2²)^{log₂log₂7}  but should have put 4^log₂7 + 2^log₂7 to maintain the same interpretation (noting that 2^{log₂log₂7} is the same as log₂7).

Also see GingerAle's post below.

Let y = 2^x then we have 4^y + 2^y = 56.      (And x = log₂y)

Now 4^y → (2²)^y → (2^y)² 

Let z = 2^y, so now we have z² + z = 56 or z² + z - 56 = 0.   (And y = log₂z)

This factorises as. (z + 8)(z - 7) = 0

Hence: z = 7,  y = log₂7, x = log₂(log₂7)

We can't have logs of negative numbers (as long as we aren't considering complex numbers) so this is the only solution.

For e^anything to be 1 we must have anything=0. Thus here we must have sinx =0.

This means x can be 0 or any integer multiple of pi.

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Yes.

If integer n is exactly divisible by 9 then it can be expressed as n = 9*m where m is another integer.

This means that n/3 = 9*m/3 → 3*m, an integer.

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