Video Transcript
Fill in the blank: two 𝑥 to the
power of six times six 𝑥 squared is equal to blank.
In this question, we’re given an
equation, and we’re asked to fill in the blank. We can see that the blank is the
entire right-hand side of this equation. So it must be equal to the
left-hand side of this equation. So to find the right-hand side of
this equation, we’re going to need to simplify the left-hand side of this
equation. And to do this, we need to notice
something. Two 𝑥 to the power of six and six
𝑥 squared are both examples of monomials. Remember, a monomial is a product
of constants and variables, where our variables are only raised to positive integer
values, which is exactly what’s happening here because our exponents of 𝑥 are two
and six, which are positive integers.
So to multiply these together, we
need to recall our rule for multiplying monomials. We know 𝑥 to the power of 𝑚
multiplied by 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of 𝑚 plus 𝑛. In other words, to multiply these
together, we just add their exponents. The first thing we’re going to want
to do is multiply our coefficients together. We want to multiply two by six. Of course, two multiplied by six is
equal to 12. So our new coefficient is going to
be 12.
Next, we need to multiply 𝑥 to the
power of six by 𝑥 to the power of two. And we’re going to do this by
adding the exponents together. 𝑥 to the power of six multiplied
by 𝑥 to the power of two is 𝑥 to the power of six plus two. And of course, we know that six
plus two is equal to eight. This gives us 12 times 𝑥 to the
power of eight. And this is our final answer. Therefore, by using the product
rule for monomials, we were able to show that two 𝑥 to the power of six times six
𝑥 squared is equal to 12𝑥 to the power of eight.
