Product of Powers Property

In math, there are a variety of operations we can do to manipulate numbers. The most common ones are addition, subtraction, multiplication, and division. There is another operation called exponentiation, which is the use of exponents or powers to multiply a number by itself n number of times. Exponents are a convenient way to express repeated multiplication, and they are commonly used in algebra, calculus, and other areas of math.

One important property of exponents is the power of a product property, which tells us how to simplify an expression that involves both multiplication and exponents. The power of a product property states that when we raise a product of two or more numbers to an exponent, we can distribute the exponent to each factor in the product and then multiply the resulting powers. In other words, ${\left(ab\right)}^n=a^n\times b^n$ .

How exponents work

So $5^2$ means 5 to the power of 2, or $5\times 5$ . Don't get confused and think that it means $5\times 2$ - with exponents, 5 is the base and 2 is the exponent, meaning the number of times you multiply 5 by itself. $5^2$ is also called 5 squared because it's the area of a square with sides of 5 units.

You can also have $5^5$ , or $5\times 5\times 5\times 5\times 5$ , or really, $5^n$ , or any number. We can find out the solution to the exponent by multiplying the factor n times. For example, $4^4=4\times 4\times 4\times 4=16\times 16=256$ .

How the product of powers property works

There is a property that helps us simplify exponents that have the same base when used in equations. That is the product of powers property. The product of powers property states that:

$a^b\times a^b=a^{b+c}$

Let's put that into action.

Say you have an equation that states $4^3\times 4^5$ .

As we now know how exponents work, we know that this problem actually means:

$\left(4\times 4\times 4\right)\times \left(4\times 4\times 4\times 4\times 4\right)$

If we were to remove the parentheses, we would have:

$4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$

Which could also be written as $4^8$ .

Now, if you look at the original exponents, you'll notice that we have 4 to the third and fifth power. The exponents are 3 and 5, which added together make 8. In other words, if you wrote the equation out this way:

$4^3\times 4^5=4^{3+5}$

You'd get:

$4^8$

This works for any exponents that are added that have the same base, whether that base is the number 90 or the term xyz.

Topics related to the Product of Powers Property

Exponents

Power of a Product Property

Power of a Product Property

Flashcards covering the Product of Powers Property

8th Grade Math Flashcards

Common Core: 8th Grade Math Flashcards

Practice tests covering the Product of Powers Property

MAP 8th Grade Math Practice Tests

8th Grade Math Practice Tests

Get help learning about the product of powers property

Tutoring is an excellent way for your student to learn all about the product of powers property. Tutoring supplements classroom learning and is done in a 1-on-1 setting that puts your student at the center of attention. Whatever your student needs to understand this and other important math concepts, their tutor can provide, whether that's slowing down and walking through each step or coming at the concepts from a different angle to help your child understand in their own way. To learn more about how tutoring can help your student improve their comprehension of the product of powers property, contact the helpful Educational Directors at Varsity Tutors today.