Unlike English or history classes, math classes tend to focus on a specific set of skills each school year. Some skills are introduced in a lower grade and then revisited in a more advanced grade. This can lead to some confusion as students learn different aspects of a skill in small doses over several different school years. One such skill is exponents, which are often introduced in middle school, revisited in Algebra 1, and then retaught in Algebra 2. An exponent is a mathematical operation that involves a base number and power. The exponent of a number is how many times you would multiply that based number times itself. Exponents can be utilized in several mathematical operations, including adding & subtracting values with exponents, and multiplying & dividing values with exponents.

### Here are the rules of dividing exponents.

The quotient property of exponents allows the division of exponents with the same base, n^{a } ÷ n^{b} = n^{(a-b)}. For example:

2^{5} ÷ 2^{2} = 2^{(5-2)} or 2^{3}

Because of the common base, the exponents can be subtracted. When dividing exponents with different coefficients, one must divide the coefficients separately and then apply the quotient property to the expression’s exponents. For example:

8a^{7} ÷ 4a^{3}^{ }

8÷4 = 2, and a^{7-3=4}^{ }

2a^{4}

^{ }

Another way to tackle different bases is to find a common number between the two digits. Consider the expression:

9^{3} ÷ 3^{2}

The number 9 can be rewritten into a 3^{2} so the coefficient in the numerator and the denominator are the same. After changing the base, one must remember to apply the original exponent to the new integer.

(3^{2})^{3} ÷ 3^{2}

The power rule allows mathematicians to raise an exponent to another exponent by multiplying them together. For instance: 3^{(2*3)} = 3^{6}. Now that the numerator is simplified, the quotient property can be used to complete to expression:

3^{6} ÷ 3^{2} = 3^{(6-2)} or 3^{4}

In addition, one might come across one or more negative exponents within a single expression. To solve this, we must understand the negative exponent rule. For every base number with a negative exponent, take the reciprocal of the base and multiply the value according to the value of the power. Consider the number: 4^{-2}. According to the negative exponent rule it would be rewritten as:

1/4^{2} or 1/(4 *4) = 1/16

Let’s apply this to the following problem: 8^{2} ÷ 2^{-2}. First, we must rewrite the expression following the negative exponent rule and the quotient property for exponents.

8^{2} ÷ (1/(2^{2}))

64 ÷ (1/4) = 64 * (4/1) = 256

### Let’s practice with a few more examples!

- 3
^{-5}3^{-2}

3^{(-5-(-2))} = 3^{(-5+2)} = 3^{-3} Subtract the exponents because they share the same base.

1/(3^{3}) = 1/9 Simplify

- 16
^{2 }÷ 2^{3}

(2^{4})^{2} ÷ 2^{3} Find a common base

2^{4×2} ÷ 2^{3} = 2^{8} ÷ 2^{3} Apply power rule

2^{8-3} = 2^{5} = 32 Apply quotient property of exponents and simplify

- 4x
^{4}2x^{2}

(4/2) x^{(4-2)} Divide coefficients and apply the quotient property of exponents

2x^{2} Simplify

With practice, students can master the rules of dividing exponents. Exponents are used extensively in science, engineering, economics, accounting, and many other disciplines. Once a student is comfortable with dividing exponents, they can be confident in moving on to more advanced topics in math.

Author: Maerie Morales