2^12-2^6/2^6-2^3 = ?

[PANJIEMING]

Hi could somebody explain how do deal with Exponents when subtracting them?

2^12-2^6/2^6-2^3 = ?

A. 2^6+2^3
B. 2^6-2^3
C. 2^9
D. 2^3
E. 2

THANKS!

[magizhan]

You dont have to subtract the exponents here. Looking at the expression we see Numerator in the form of a^2-b^2 which can be written as (a+b)(a-b) with a=2^6 and b=2^3. (a-b) in the denominator gets cancelled leaving a+b alone. Hence A is the answer.

[thephoenix]

2^12-2^6/2^6-2^3 =2^6(2^6-1)/2^3(2^3-1)=2^3((2^3)^2-1))/(2^3-1)[using a^2-b^2(9a-b)(a+b)]
2^3[(2^3-1)(2^3+1)]/(2^3-1)=2^3(2^3+1)=2^6+2^3=A

[Stuart@KaplanGMAT]

Hi could somebody explain how do deal with Exponents when subtracting them?

2^12-2^6/2^6-2^3 = ?

A. 2^6+2^3
B. 2^6-2^3
C. 2^9
D. 2^3
E. 2

THANKS!

There's no simple way to add and subtract exponents; the only time we can do so is when both the base and the power are the same in each term. When that happens, we simply add or subtract the coefficients.

For example:

5(x^2) + 8(x^2) = (5+8)x^2 = 13(x^2)

However, if we had:

5(x^3) + 8(x^2), there's no easy way to combine those terms.

So, if we want to add or subtract, we need to equalize the powers; we do so via factoring out.

To do so, we use the multiplicative exponent rule:

x^a * x^b = x^(a+b)

Let's look at the numerator from this question:

2^12 - 2^6

we can express the first term as:

2^6 * 2^6, since

2^6 * 2^6 = 2^(6+6) = 2^12

Simplifying, we get:

64(2^6)

Substituting that new term into the original expression:

64(2^6) - 2^6

= 63(2^6)

We could then express denominator as a power of 6 as well.

All of that aside, that's not how we want to attack this particular question (the math gets super ugly); the method suggested by magizhan is vastly superior.

One key to GMAT success is pattern recognition; a very commonly tested pattern in math is the difference of squares.

Whenever you see two perfect squares separated by a minus sign, think in terms of:

a^2 - b^2 = (a+b)(a-b)

and how you can use that patter to solve the question.

In this case, we can express the numerator as:

(root(2^12) + root(2^6)) * (root(2^12) - root(2^6))

= (2^6 + 2^3) * (2^6 - 2^3)

plugging that into the full expression, we now have:

(2^6 + 2^3) * (2^6 - 2^3) / (2^6 - 2^3)

= 2^6 + 2^3