Exponents

The main test with exponents is to know when to perform the core three math operations to exponents:

Three Main Exponent Operations

• Addition:

Exponents are added together when we those exponents share the same base number and that number is multiplied by itself

x² • x⁵ = x^{5 + 2 = 7} = x⁷

• Subtraction:

Exponents are subtracted together when we those exponents share the same base number and that number is divided by itself. The number in the denominator is subtracted from the number in the numerator.

x⁵/x² = x^{5 – 2} =x³

• Multiplication

Exponents are multiplied by one another when a base number and its exponent are raised to another power (when an exponent-base pair has an exponent).

(x⁵)³ = x^5 • 3  = x¹⁵

There are also three special exponents that students are expected to know. Instead of using operators, these are types of exponents. Students must know the different ways to express these exponents, and how that can then affect the operations listed above:

Special Exponents

• Negative Exponents

Negative exponents signify that a number is being taken to the inverse of the given exponent.The exponent expression gets moved from the numerator to the denominator.

x⁻⁴ = 1/x⁴

A key takeaway is that a negative exponent doesn’t lead to negative numbers. The negative here is about inversion, not positive/negative.

• Fractional Exponents

Fractional Exponents are another way to express roots.The root that students are generally most familiar with is the square root. Square roots are thought to be the default when the root operator is used. But this could also be written as

²√

if a student wanted to be clear that it’s a square root (or root 2). This makes then makes it easier to understand the rule that:

x^1/2= ²√x

The denominator becomes the root power.

x^2/3= ³√x²

If there is a number other than one in the numerator, that exponent stays with the base number under the radical, even when we switch to the root expression.

It’s important to remember: Fractional exponents do not lead to the base switching from numerator to denominator (as happens with negative exponents)

• Zeroed Exponent

Any number taken to the 0 power equals 1.

x⁰= 1 3⁰ = 1 ( − 13)⁰ = 1