Question Video: Simplifying Algebraic Expressions Using Laws of Exponents Mathematics

Video Transcript

Simplify seven 𝑥 to the fourth power times eight 𝑥 to the seventh power.

There are two things we know about multiplication. It is associative and commutative. The associative property tells us that the product will be the same no matter how the numbers are grouped. And the commutative property tells us that the order in which we perform the calculation really doesn’t matter. And so, we’re going to begin by sort of unsimplifying each term in our expression. When we do, we get seven times 𝑥 to the fourth power times eight times 𝑥 to the seventh power. We’re now going to switch around 𝑥 to the fourth power and eight. And so, we see that our expression now becomes seven times eight times 𝑥 to the fourth power times 𝑥 to the seventh power. And in fact, we can work out the numerical part. We know that seven times eight is 56, which means that this becomes 56 times 𝑥 to the fourth power times 𝑥 to the seventh power.

But what do we do with these algebraic parts? Well, we have a rule for multiplying exponential terms. As long as the base is the same, we add the exponents. So, 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. This in turn means that 𝑥 to the fourth power times 𝑥 to the seventh power is 𝑥 to the power of four plus seven. And since four plus seven is 11, this becomes 𝑥 to the 11th power. Now, in fact, we know that we don’t really want to include that multiplication symbol. And so, we simplify this further to get 56𝑥 to the 11th power. And so, when we simplify seven 𝑥 to the fourth power times eight 𝑥 to the seventh power, we get 56𝑥 to the 11th power.