The most important thing that I learned in Math 10 this week was how to take equations with negative exponents and convert them to positive exponents. I chose this topic because I had not previously known anything about negative exponents, and learning how to evaluate a negative exponent is very important because any kind of math problems in the future could involve them and they’re very different from how you evaluate positive exponents.

Negative exponents of a number follow the same pattern as its positive exponents except flipped to the other side of the fraction. To understand this you have to know that all integers are fractions, but they don’t present as a fraction because the 1 divides into the number; for example, 2 is equal to . With negative exponents, the result of the power is the same number but flipped because a base with a negative exponent is showing that it is on the wrong side of the fraction and we can move it to the other side to have it in its positive form, which presents it more clearly and simplified. Like positive exponents, the negative exponent only affects its base, so that is the only part of the equation that you would move from the numerator position to the denominator position or vice versa, removing the negative; in a larger equation there could be multiple negative exponents to move.

A simple example of how this works is

If which is really , follows the same pattern and is therefore the same answer flipped. Because the -2 is in the numerator position, you switch it to the denominator position and drop the negative, making it .

In a more complicated equation with more elements, you incorporate this law and use it as well as the other exponent laws: product law in which you add exponents on the same base to multiply (), quotient law in which you subtract exponents on the same base to divide (), and power law in which you multiply the exponent outside the brackets with what is inside the brackets ().

When you have an equation in which more than one of these laws apply, you can do them in any order. In this example, I chose to do the negative exponent law first, as it is the subject of this blog post.

Step 1. Find all of the bases with negative exponents and move them from the numerator to the denominator or vice versa.

Step 2. Remove the negative from the exponents.

Step 3. Further simplify the equation using the other exponent laws.

Product law:

Quotient law: