# Just Do Something

“I don’t even know how to start” is a frequent refrain overheard by SAT math students as they tackle problems on the SAT. Half  the battle on SAT math sections is being sure you are answering the right question. Your algebra teacher asks you to “solve for $x$” in an equation; the SAT demands, “solve for $3x-y$ .” Let’s look at the following example, taken from a College Board practice exam.

D) The value cannot be determined from the information given.

The above question stumps lots of students. For starters, the two given expressions seem unrelated. The linear equation $3x-y=12$ doesn’t seem to have much in common with the expression  $\frac{8^x}{2^y}$  except they both share the same variables. Where do we begin? The bottom line: Just Do Something.

Some students suggest that we start by isolating the variable $y$ in the linear equation. Okay, let’s see what happens:$3x-y=12$
$-y=-3x+12$
$y=3x-12$

At this point, we could substitute $3x-12$ for the variable and proceed but it seems as though that will make things more complicated, not easier. Another approach might be faster.

Let’s try to simplify the expression $\frac{8^x}{2^y}$ instead. At first glance, it may seem there isn’t a lot to do here. But, let’s remember rules when dividing with exponents: if the bases are the same, we can subtract the powers. On SAT math, if the bases don’t match, try to write the larger base as a power of the smaller. Here, the bases differ but we can write 8 as a power of 2, then proceed:

$\frac{8^x}{2^y}=$

$\frac{(2^{3})^{x}}{2^{y}}=$

$\frac{2^{3x}}{2^{y}}=$

$2^{3x-y}$

Does anything look familiar? Indeed, $3x-y$ jumps out at us! We know from the given linear expression that $3x-y=12$ so we can substitute 12 for $3x-y$. Hence, our answer is $2^{12}$. Voila!

The correct answer is Choice (A).

Here’s another example, taken from a College Board practice exam:

A)  8

B)  14

C)  16

D)  18

Once again it’s not immediately clear how the equations $\frac{x^{a^{2}}}{x^{b^{2}}}=x^{16}$and $a+b=2$ are related. My advice: Just Do Something. In this case, we can simplify the left side of the equation $\frac{x^{a^{2}}}{x^{b^{2}}}=x^{16}$ since the numerator and denominator of the fraction have the same base. Applying rules of exponents, we can simplify as follows:

$\frac{x^{a^{2}}}{x^{b^{2}}}=x^{16}$

$x^{a^{2}-b^{2}}=x^{16}$

To simplify further, we can factor the difference of squares ${a^{2}-b^{2}}=(a-b)(a+b)$ and  substitute:
$x^{(a-b)(a+b)}=x^{16}$

Setting the exponents equal to each other, we obtain:

$(a-b)(a+b)=16$

Now we’re getting somewhere! Hopefully, we see how the equation $a+b=2$ fits into the problem! We can substitute 2 for $a+b$ and we have

$(a-b)\cdot&space;2=16$

It’s evident that the expression $a-b$ must be equal to 8.

The correct answer is Choice (A).

The bottom line: on SAT math, sometimes you just need to do some simplifying or applying of math rules before you see how the problem can be solved. It takes some confidence and some practice but this approach can help you tremendously on the SAT!