Most Properties are True in Both Directions

In this very short article, we’ll examine a common phenomenon that can prevent you from maximizing your quant score.

The phenomenon: students often fail to view mathematical properties in both directions.

Consider, for example, the Product Law for exponents: (b^x)(b^y) = b^(x+y)

This is a pretty straightforward property, and when students encounter an expression such as (x^k)(x^3), they're likely to rewrite it as x^(k+3)

These same students, however, often fail to see that we can take the expression x^(k+3) and rewrite it as (x^k)(x^3).

These students view the Product Law in one direction. That is, if two powers have the same base, we can rewrite the product of those powers as the base raised to the SUM of the exponents. These students fail to see that a power with a SUM in the exponents can be rewritten as the product of two powers.

This phenomenon happens a lot with questions involving exponential properties, but it also occurs elsewhere. For example, most students know that, if we have a right triangle, then a^2 + b^2 = c^2 (where a, b and c represent the three side lengths). However, the reverse is also true: if we have a triangle such that a^2 + b^2 = c^2, then that triangle is a right triangle.

Important: Not all mathematical properties are true in both directions. For example, if quadrilateral ABCD is a square, then it's also true that quadrilateral ABCD is a rhombus. However, the reverse is not necessarily true. That is, if quadrilateral ABCD is a rhombus, we can't conclude that quadrilateral ABCD is a square.

So, as you learn/relearn various mathematical properties, be sure to understand whether those properties are true in both directions. If a property is true in both directions, be sure to remember this on test day.

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