Question Video: Simplifying an Expression Using the Product Rule of Exponents Mathematics

Video Transcript

Fill in the blank: two squared multiplied by 𝑥 cubed times two to the power of seven multiplied by 𝑥 to the power of five is equal to two to the power of nine multiplied by 𝑥 to the power of blank.

In this question, we’re given an equation and we’re told to fill in the blank in our equation. We can see the blank in our equation is the power of 𝑥 on the right-hand side of the equation. So we want to find the power of 𝑥 which makes this equation true. To do this, we need to make sure both sides of our equation are equal. So let’s take a look at the left-hand side of this equation. We can see we’re multiplying four terms together. In fact, we can see something interesting. All of these are raised to positive integer values. And because they’re raised to positive integer values, this means this is the product of monomials. So we can simplify this by using the product rule for monomials.

Remember, this tells us that 𝑥 to the power of 𝑚 multiplied by 𝑥 to the power of 𝑛 will be equal to 𝑥 to the power of 𝑚 plus 𝑛. We just need to add the powers together. So let’s try applying this to the left-hand side of our equation. First, it might help to recall that we can multiply numbers in any order. It won’t change its value. So we can switch the second and third factor in this expression around. So we can rewrite this expression. It’s two squared multiplied by two to the power of seven times 𝑥 cubed times 𝑥 to the power of five.

Now, we can simplify two squared multiplied by two to the power of seven. All we need to do is add their powers together. Two squared multiplied by two to the power of seven is just equal to two to the power of two plus seven. And we can do exactly the same for 𝑥 cubed multiplied by 𝑥 to the power five. To multiply these, we just add their powers together. We get 𝑥 to the power of three plus five. Now, we can simplify this. In the power of two, two plus seven is equal to nine. So this simplifies to give us two to the power of nine. And in the power of 𝑥, three plus five is equal to eight. So we get 𝑥 to the power of eight. And now we can see this is exactly the form given to us in the right-hand side of our equation, where the power of 𝑥 is equal to eight, so the value of the blank should be equal to eight.

Therefore, by using the product rule for monomials, we were able to show for two squared times 𝑥 cubed multiplied by two to the power of seven times 𝑥 to the power of five to be equal to two to the power of nine times 𝑥 to the power of blank, the blank must be equal to eight.