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What Are the Five Main Exponent Properties?

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  • 0:30 Product of Powers
  • 1:10 Power to a Power
  • 1:50 Quotient of Powers
  • 2:53 Power of a Product
  • 3:40 Power of a Quotient
  • 4:41 Lesson Summary
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Lesson Transcript
Instructor: Kathryn Maloney

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

We'll look at the five important exponent properties and an example of each. You can think of them as the order of operations for exponents. Learn how to handle math problems with exponents here!

Understanding the Five Exponent Properties

We are going to talk about five exponent properties. You can think about them as the order of operations for exponents. Just like the order of operations, you need to memorize these operations to be successful. The five exponent properties are:

The Quotient of Powers property
Quotient of Powers Property

  • Product of Powers
  • Power to a Power
  • Quotient of Powers
  • Power of a Product
  • Power of a Quotient

Let's look at the first one.

Product of Powers

Here's the formula: (x^a)(x^b) = x^(a + b). When you multiply exponentials with the same base (notice that x and x are the same base), add their exponents (or powers).

Let me show you how that works. Let's say I have (x^2)(x^3). Well, x^2 is x times x, and x^3 is x times x times x. When we add all those xs up, we get x^5, which is the same thing as adding 3 + 2.

Power to a Power

The Power of a Product property
Power of a Product Property

We can see from the formula we have (x^a)^b. When you have a power to a power, you multiply the exponents (or powers). Let me show you how this one works.

If I have (x^2)^4, which would be x^2 multiplied four times, or x^2 times x^2 times x^2 times x^2. Once again, we add all the exponents and get x^8, and x^8 is the same as x^(2 * 4), which is 8. Not too bad, right?

Quotient of Powers

Remember, 'quotient' means 'division'.' The formula says (x^a) / (x^b) = x^(a - b). Basically, when you divide exponentials with the same base, you subtract the exponent (or powers).

Let me show you how this one works. Let's say I had (x^4) / (x^3). In the top (or numerator), we have x times x times x times x. In the bottom (or denominator), we have x times x times x. Hopefully, you remember that x divided by x is 1, so the xs cancel. So, x divided by x is 1, x divided by x is 1, and x divided by x is 1. So, when we cancel them, what are we left with? That's right: x^1, or just x. So (x^4) / (x^3) is just x^(4 - 3), which is x^1.

The Power of a Quotient Property
Power of a Quotient Property

Power of a Product

The formula says (xy)^a = (x^a) and (y^a). When you have a product of a power, you give each base its own exponent. Think about it as distribution putting the exponent with each base.

Let me show you how this one works. Let's say we had (xy)^2. That means we take xy and multiply it twice that means xy times xy. Well, that would give us two xs, or x^2, and two ys, or y^2. That is the same as if I distributed 2 to the x, getting x^2, and 2 to the y, getting y^2.

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