Translational motion test, #5

Is there an easier way to do this problem than what is written as the explanation?  The explanation is rather difficult to understand.
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This is one of the most asked about questions in the whole Kaplan curriculum.  Here’s my explanation:

This question reads:  “Cells can only absorb nutrients through their surfaces.  In order for a cell to absorb nutrients most efficiently, it will grow to a maximum size and then divide.  If the total volume remains constant when a cell divides, then the ratio of the total surface area of the daughter cells to that of the original cell is:”

Let’s start with defining a few variables here.  According to our passage, “each animal has a characteristic length L…its area is proportional to L²…and its volume is proportional to L³.”

Let’s also define the subscript “1” to mean the cell before it’s divided, and “2” to represent each cell after it’s divided.

The total volume here is constant – the one cell before division has the same volume as adding together the two cells after division.  This gives us the equation that L₁³=2L₂³.  Let’s take the cube-root of both sides.  Well, the cube root is the same as taking each side to the 1/3 power.  So we would have:

(L₁³)^1/3=(2L₂³)^1/3.  When you have an exponent raised to an exponent, we multiply their values.  This means that this simplifies out to L₁=2^1/3L₂.  The 2^1/3 is because we have to distribute the exponent to both values in the parentheses.

From here, we can figure out the answer to our question.  It asks us to find the total surface area of the daughter cells, as compared to the area of the original cell, or

2L₂²:L₁².  Remember that there are TWO daughter cells, so that’s why we have the two here.  Well, we know how L₁ and L₂ are related, so we can say that

2L₂²:L₁² can be represented as 2L₂²:(2^1/3L₂)² – we just substituted in our value for L₁ from that volume calculation before.  Distributing our exponent on the right-hand side, we have 2L₂²:2^2/3L₂².  Let’s divide both sides by L₂².  That gives use 2:2^2/3.  But… that doesn’t quite look like any of the answers here.  Since most of the answers have 1 on the right-hand side, let’s divide both sides by 2^2/3.  That gives us:

[2/2^2/3]:[2^2/3/2^2/3].  When we divide two items that have the same root and different exponents, we simply subtract the exponents.  That means that we have [2^1-2/3]:1, or 2^1/3:1.  That corresponds to answer choice (C).  If you’re having difficulty with exponents, I encourage you to go on to Google and type in “Exponent Practice.”  You want to be able to work with exponents very easily on the MCAT.  The four big rules are:

(X^A)(X^B) = X^A+B

(X^A)/(X^B) = X^A-B

(X^A)^B = X^AB

nth root of X = X^1/n